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     "source": [
      "# MTH 9879 Market Microstructure Models,    Spring 2015 \n",
      "\n",
      "## Lecture 6: Variance and covariance estimation\n",
      "\n",
      "Tai-Ho Wang    \n",
      "Department of Mathematics   \n",
      "  \n",
      "  \n",
      "<h2><img src=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2014/09/BCCUNYstacked_BLK.jpg\" align = \"center\" width=160></h2>  \n",
      "\n",
      "$$\n",
      "\\newcommand{\\bea}{\\begin{eqnarray}}\n",
      "\\newcommand{\\eea}{\\end{eqnarray}}\n",
      "\\newcommand{\\supp}{\\mathrm{supp}}\n",
      "\\newcommand{\\F}{\\mathcal{F} }\n",
      "\\newcommand{\\cF}{\\mathcal{F} }\n",
      "\\newcommand{\\E}{\\mathbb{E} }\n",
      "\\newcommand{\\Eof}[1]{\\mathbb{E}\\left[ #1 \\right]}\n",
      "\\def\\Cov{{ \\mbox{Cov} }}\n",
      "\\def\\Var{{ \\mbox{Var} }}\n",
      "\\newcommand{\\1}{\\mathbf{1} }\n",
      "\\newcommand{\\p}{\\partial}\n",
      "\\newcommand{\\PP}{\\mathbb{P} }\n",
      "\\newcommand{\\Pof}[1]{\\mathbb{P}\\left[ #1 \\right]}\n",
      "\\newcommand{\\QQ}{\\mathbb{Q} }\n",
      "\\newcommand{\\R}{\\mathbb{R} }\n",
      "\\newcommand{\\DD}{\\mathbb{D} }\n",
      "\\newcommand{\\HH}{\\mathbb{H} }\n",
      "\\newcommand{\\spn}{\\mathrm{span} }\n",
      "\\newcommand{\\cov}{\\mathrm{cov} }\n",
      "\\newcommand{\\HS}{\\mathcal{L}_{\\mathrm{HS}} }\n",
      "\\newcommand{\\Hess}{\\mathrm{Hess} }\n",
      "\\newcommand{\\trace}{\\mathrm{trace} }\n",
      "\\newcommand{\\LL}{\\mathcal{L} }\n",
      "\\newcommand{\\s}{\\mathcal{S} }\n",
      "\\newcommand{\\ee}{\\mathcal{E} }\n",
      "\\newcommand{\\ff}{\\mathcal{F} }\n",
      "\\newcommand{\\hh}{\\mathcal{H} }\n",
      "\\newcommand{\\bb}{\\mathcal{B} }\n",
      "\\newcommand{\\dd}{\\mathcal{D} }\n",
      "\\newcommand{\\g}{\\mathcal{G} }\n",
      "\\newcommand{\\half}{\\frac{1}{2} }\n",
      "\\newcommand{\\T}{\\mathcal{T} }\n",
      "\\newcommand{\\bit}{\\begin{itemize}}\n",
      "\\newcommand{\\eit}{\\end{itemize}}\n",
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      "\\newcommand{\\eeq}{\\end{equation}}\n",
      "\\newcommand{\\tr}{\\mbox{tr}}\n",
      "\\newcommand{\\underbar}{\\underline}\n",
      "$$\n"
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     "source": [
      "## Outline of Lecture 6\n",
      "\n",
      "-   Uses of volatility estimates\n",
      "\n",
      "-   Market microstructure bias\n",
      "\n",
      "-   A survey of estimation and forecasting algorithms\n",
      "\n",
      "-   Experimental results\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
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     },
     "source": [
      "### Motivation\n",
      "\n",
      "-   We often need to estimate in-sample volatility. Typically we need a\n",
      "    good volatility estimate to reduce errors in estimating parameters\n",
      "    of:\n",
      "\n",
      "    -   Market impact models\n",
      "\n",
      "    -   Limit order fill models\n",
      "\n",
      "    -   Volatility forecasting models: without good in-sample volatility\n",
      "        estimates, how can we even assess the quality of volatility\n",
      "        forecasts?\n",
      "\n",
      "-   Also, it has been shown that the performance of volatility\n",
      "    forecasting models may be improved if they use better estimates of\n",
      "    realized variance.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Uses of volatility forecasts\n",
      "\n",
      "-   Option valuation\n",
      "\n",
      "-   Risk estimation\n",
      "\n",
      "-   Order fill probability\n",
      "\n",
      "-   For the first two of these, we need to estimate the width of the\n",
      "    distribution of relatively long-timescale returns.\n",
      "\n",
      "-   For the order fill probability, we need a volatility to estimate the\n",
      "    first passage time density. Given that the underlying stochastic\n",
      "    process is not Brownian motion, there is no a priori reason why the\n",
      "    volatility numbers required for these quite different computations\n",
      "    should be the same.\n",
      "\n",
      "-   We will focus on measuring the width of the distribution of returns.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Choice of sampling scheme\n",
      "\n",
      "It is important to sample in transaction time rather than in business\n",
      "time or calendar time.\n",
      "\n",
      "-   In transaction time, empirically observed returns are MA(1).\n",
      "\n",
      "-   In calendar time, with varying intensity, empirically observed\n",
      "    returns are ARIMA.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### TAQ data cleaning\n",
      "\n",
      "Data cleaning is critical. Here follows a recipe for TAQ cleaning due to\n",
      "[Barndorff-Nielsen et al.]<sup id=\"cite_ref-BNS\" class=\"reference\"><a href=\"#cite_note-BNS\"><span>[</span>1<span>]</span></a></sup> and implemented in the R-package RTAQ:\n",
      "\n",
      "<span>**All data**</span>\n",
      "\n",
      "- Delete entries with a timestamp outside the 9:30am\u20134pm window when the exchange is open.\n",
      "- Delete entries with a bid, ask or transaction price equal to zero.\n",
      "- Retain entries originating from a single exchange (NYSE in our application). Delete other entries.\n",
      "\n",
      "<span>**Quote data only**</span>\n",
      "\n",
      "- When multiple quotes have the same time stamp, we replace all these with a single entry with the median bid and median ask price.\n",
      "- Delete entries for which the spread is negative.\n",
      "- Delete entries for which the spread is more that 50 times the median spread on that day.\n",
      "- Delete entries for which the mid-quote deviated by more than 10 mean absolute deviations from a rolling centered median (excluding the observation under consideration) of 50 observations (25 observations before and 25 after).\n",
      "\n",
      "<span>**Trade data only**</span>\n",
      "\n",
      "- Delete entries with corrected trades. (Trades with a Correction Indicator, CORR $\\neq$ 0).\n",
      "- Delete entries with abnormal Sale Condition. (Trades where COND has a letter code, except for E and F). See the TAQ 3 Users Guide for additional details about sale conditions.\n",
      "- If multiple transactions have the same time stamp, use the median price.\n",
      "- Delete entries with prices that are above the ask plus the bid\u2013ask spread. Similar for entries with prices below the bid minus the bid\u2013ask spread.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Realized Variance\n",
      "\n",
      "Given a set of tick data, how can we measure variance? The na\u00efve answer\n",
      "would be to compute the statistic\n",
      "\n",
      "$$\\frac{1}{T}\\,\\sum_i^T\\,\\log \\left( \\frac{S_i}{S_{i-1}}\\right)^2$$ \n",
      "\n",
      "where the $S_i$ are\n",
      "successive prices in the dataset.\n",
      "\n",
      "-   This estimator is called the Realized Variance (RV) estimator\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### An exact relation\n",
      "\n",
      "-   The following relation is exact:\n",
      "    $$(p_T-p_0)^2 = \\sum_{i = 1}^T \\,r_i^2 + 2\\sum_{k = 1}^{T-1}\\,\\gamma(k)$$\n",
      "    where $r_i = p_i-p_{i-1}$ and\n",
      "    $\\gamma(k) = \\sum_{i = 1}^{T-k}r_i \\,r_{i+k}$.\n",
      "\n",
      "\n",
      "-   So if returns are serially uncorrelated then an unbiased and\n",
      "    efficient estimate of the daily realized variance can be obtained as\n",
      "    the sum of squared intra-day returns (RV).\n",
      "\n",
      "\n",
      "-   However, intra-day returns sampled at the highest frequency will\n",
      "    generally exhibit serial correlation thereby invalidating RV as a\n",
      "    reliable variance estimator.\n",
      "\n",
      "\n",
      "-   The estimators we discuss below are all motivated by this reasoning\n",
      "    and aim to provide improved measures of the realized variance.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Microstructure noise\n",
      "\n",
      "In the limit of very high sampling frequency, RV picks up mainly market\n",
      "microstructure noise. To see this, suppose that the observed price $Y_t$\n",
      "is given by\n",
      "\n",
      "$$Y_t =X_t +\\epsilon_t$$\n",
      "\n",
      "where $X_t$ is the value of the\n",
      "underlying (log-)price process of interest and $\\epsilon_t$ is a random\n",
      "market microstructure-related noise term. Suppose we sample the price\n",
      "series $n+1$ times (so that there are $n$ price changes) in the time\n",
      "interval $T$. Then as $n\\to\\infty$,\n",
      "\n",
      "$$[Y,Y] :=\\sum^T\\,(Y_i-Y_{i-1})^2\\sim[X,X]+2\\,n\\,Var[\\epsilon]=\\sigma^2+2\\,n\\,Var[\\epsilon]$$\n",
      "\n",
      "where $\\sigma^2$ is the (true) variance for the whole period.\n",
      "\n",
      "-   The estimated variance $[Y,Y]$ increases without limit as\n",
      "    $n \\to \\infty$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### BAC trades"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "BAC data from 04-May-2012."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%load_ext rmagic"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The rmagic extension is already loaded. To reload it, use:\n",
        "  %reload_ext rmagic\n"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%R\n",
      "download.file(url=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2015/02/tq.zip\", destfile=\"tq.zip\")\n",
      "unzip(zipfile=\"tq.zip\")\n",
      "download.file(url=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2015/03/RvEstimators.R.zip\", destfile=\"RvEstimators.R.zip\")\n",
      "unzip(zipfile=\"RvEstimators.R.zip\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "trying URL 'http://mfe.baruch.cuny.edu/wp-content/uploads/2015/02/tq.zip'\n",
        "Content type 'application/zip' length 695952 bytes (679 Kb)\n",
        "opened URL\n",
        "==================================================\n",
        "downloaded 679 Kb\n",
        "\n",
        "trying URL 'http://mfe.baruch.cuny.edu/wp-content/uploads/2015/03/RvEstimators.R.zip'\n",
        "Content type 'application/zip' length 2406 bytes\n",
        "opened URL\n",
        "==================================================\n",
        "downloaded 2406 bytes\n",
        "\n"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%R\n",
      "library(highfrequency)\n",
      "load(\"tq.rData\")\n",
      "source(\"RvEstimators.R\") # Code mostly due to Roel Oomen"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%R\n",
      "\n",
      "px <- tqBAC$PRICE\n",
      "p <- as.numeric(px)\n",
      "\n",
      "#----------------------------------------------------------------------------------\n",
      "# Plot time series of trades\n",
      "par(mar=c(5, 4, 1, 2) + 0.1)\n",
      "par(mfrow=c(1,2),cex=0.6)\n",
      "plot(p,col=\"red\",type=\"l\",ylab=\"Trade price\",xlab=\"Trade #\")\n",
      "plot(100:200,p[100:200],col=\"red\",type=\"l\",ylab=\"Trade price\",xlab=\"Trade #\")\n",
      "par(mfrow=c(1,1))"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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L2G049HxOh44OVFaipwcFBaZtU243mtRcO+G7rXbQ8gIuyiBVAq3poGNTHOJB\ntxHloB9//PFgMHjcccfdc889dgVE0sXnS33Bb2/hzfO5stLkyVxKB33okHaXRCWBAILByG1Lc9Ay\ncjN+VYrjiy+kySlKRA7aeykOv98P4MiRI42mz1ZKB3cuJmYayWqoWJo+5V146FA79HxOh3A40gPL\nLJTmtLnZqIRDUFWFL7+UbltRZqd00Pjq5FSNZMrErtcF71ZxNDQ07N27d9GiRStWrLArIC8JRCZI\n3wJ710Q74nw2F58P5eUmO2jlIOGXXyYt0N3d1gq0EATZQStTHHoJwNxcdHZ6o4oj6uIzduzYO++8\ns6enp1LMbbcXKnVqJCW4qhdnuGLEYpx1PqdPezsGDUJFBVpazNys3I4uLw9NTRgxIs7rhw6NEmgg\nEykO2UErdbapCcOHa28hHPagg7722msBFBQU3HzzzTbFQ6LJzFVKluniYi9dF712PosuGaanOJQ5\n6ObmOFVDAKqq0Nws3c6AQAthVeagZZ3VHCEEpI8TO0jodgfd3d09fPhwAC3mXqLTx0Oq4Sy8m9+A\nk8/n1BCTsCsqsGuXmZtV5aATEWjRTxxfiaCQTrNQpThEpZ0yB60U6NhZKvjqguEJBx0l0Ndee+2/\n//u/h8Ph6667zq6AIojyGoHBpS9jq77bgjxcrkdxceob9/vVJ7G3cNb5nD5iEralOegEBXrzZklD\nxfljbqWmSqDFoiq9vRqDhPv24aijtLcAzwn0jh07xo4de++999oYDcrKIrpcXw8A48cjGMQNN6Ch\nQfstS5finHNw2mkZijCTbN+OK67A979v9JraWlx4oXTb50NOThJ2Zs4clJXh4EEMH47ZsxEOIy/P\n5PymfTjifDYX2UGbnuKQte/AgfgCPXQo9uyR5K+oCH6/yQKtSnHIDjo2xfHhhzjnHO0twHMCvWHD\nhg0bNpx66qni7h133GFDOOXlCIXQ0oJnn5Wuk6eeKv159EZsxfrqnkScT//yL0avGTkSTz+d4vZ/\n+MPI7d/9DgD+8Q/85Ccpbs1hOOJ8Nhc5B22dg+7oiC/QQ4Zg3z4pt1BYaP5EJ80Uh1guFtEOetcu\njBunsYWcHPh86jpot+egf/jDH77zzjszZ84cM2aMffEAANra8NZb0m25F0dLC4YMsTEo4i4cdD6b\nRUsLamutddCI11oAQGkpDhzAhAkAUFRkfrkgxz4AACAASURBVIJR00EHAmoHPTCAcFij3kOQm+vB\nqd5jxozZtGmTuG2/41DmoMFxQpI0yZ7PH3300erVq1taWsaNG7dgwYIRcQvOMoxYLbCwMNKmxhSU\nDtrnQ1lZ/LeEQlICIRAwX6A1c9C5ueoyu8ZG1NXpbiQvz4MzCadPn75t27aPP/54+vTpNsWjQO4H\nTTKJhw57sufzihUr5s+fX1VVNXPmzBtuuMHi6JJHXnzPXJQOurQ0oZTFkCEWCrSmg46dqPLee5gy\nRXcjnnTQ99xzz7PPPuvz+S644AL7NVr8jkukDkzltUnK+Hz47DO7gzCNZM/n8vLympqa7u7uKVOm\n5JtbOmYKxmudpIzSQSe4/aqqzDnoQYPQ1ISKCrWD/uADnHKK7kby8rwxSBjloEeOHLl///6mpqbh\nmvNzHAuzHybioYOZ7Pl81llnzZs374orrti6deuiRYusDi9pDh2S8g9+v5k/dJQOOlmBLiw0s6+e\nQM9BiyDlz/7++0YOOjbF4fZBQgBFRUUrV64E4Pf7r7zySgAPPfRQRsMRfZGUrtnTMyniwEXN0yPZ\n83nSpElf//rXi4qKVq9efcEFF0yePDlDgSZIKCTNxS8vx6FDpqU7lA46biMOwdChEX9qen9qzRy0\nXGYns3+/0az04mJ13wIPCPRPf/pTu+KIQtPEecjZkcyQ7Pm8fPnyyy67rL6+fuPGjT/60Y/OPPNM\niwJLEflSXVaGtjbTBFo51TtBgR4yJOJPE3xL4uTmRmmrykEDaGvD9u1xnPugQepH3NkU03nN2kMh\ntLVhYEBa8grxLKQLD7pz0Stayg78fn9JSYnP5ysoKPDH9I1qbGy85ZZb5LvdJpZS9PbikUcwf77u\nC376U7z2WiStUVWFhoaocovrrsPppwPA++/jj3/EqFGYOzfy7M9+hmuvBYB163DSSaipQTCIK65A\nfz8AvPUWli4FgMJCVFUlFPDw4ZFv5QsvJPYhE6awMMosl5bij3/EG2/g+uulR379ayxYAOPfN8ZX\nrz/+Ecccg7Fj0d+PX/0KV1+t+8rVq3HppeancRLGeSuq9PbiwAEMDEi/R+TzQNZrYh3eapaU7Pm8\naNGidevWrVmzpqGhYfbs2apnR48evXr1avnuiy++aFqgHR34wx+MBPqFF7BmTWQt1zvvjFo68rnn\n8O67kkBfcgkWL8bDD0cEOhzGk09KAv3aaxgyBDU1aGlBKATxceTSuq99DQlOvPzBDyLnienideaZ\nmDkzcnf0aGzdinA4kkuZOhUvvxxnxO+PfzR6dssW5OVh7Fh0dOD3vzcS6PXrce65GmsCZAr3rKjC\nUg2SJMmezxMnTlyyZMkJJ5zwxBNPxFZx+P3+cgU+E8cG+vvjpEf7+lBVFakby8+XmnKIf2VlkhcG\nUFqK2bPR1RV5bygU2Xh/v9TdJRhEaWnk7TIJqm1OjoUrpfn96sqQsjKUl6t/3hmX2Rh/kGBQqiXv\n7o7T7qa3196FsqKOcrIrUOzcufOhhx764osvqqurr7rqqnGa0y5TIykf5yHTR0zENSuq9PVFFFYT\nzXVDZGIXe62pweefY+RI9caVAu3pPllxCAYlw9fTE0ege3rMX0Y9GdJaUWXJkiVz5869//77586d\nu1SksdJEmcin7IojYF/+y+0kez4vXLhw8+bN4vasWbOsDC2auA7amLw8tb4fdxzefVdj4319FGjA\nTQ46ItDXX3/9o48+WlZWNnTo0GeeeSahN/v9xcXFgUBADK2YGdfAgHTOyZvNWr1WlXNaijtHujVJ\n4Xzu6OjYt2/fsmXLtm3bZnV4UcR10MbEOmilQNNBx5KUg3aIQK9cufLgwYNXXnnllVde2ZZYu6yl\nS5euWrVq7ty5q1atuvXWW00LKhxGOMxRQQkrJo9lASmczz6fr76+vqGhYf369RdffLHVEUagg84M\nsv9wj4OOJLYKCwvb2tqOPvpoAPv370/kzQUFBWPHjl2wYMF99923Z8+eiRMnmhCROIKx6uwVZ5c0\nQ4faHYErSeF8njZtGoCqqqpMTyNM30GLt8uTvEaMgPyR6aBlRJ+pQCDKQcvlkvv2obpaPRTZ2+ug\nHPQ111wze/bsiy66SCzmFpfly5dPnTq1vr5+xYoVa9eutSZCQlIk2fN54cKFVoekTfoOWrxdOZZY\nWChpMR20jJBm6KQ47rwTH3+sfotzHDSAuro6MXvq0UcfPeuss+K+2biwPxgMbtmyRb4bSrB7wMGD\n+OCDyKqUMipPffAg3n0XtbXYs8ez5tqWMyPuwTx0CH//e+TXYl2ddtN0B5Ds+WwbZjlo5fJvU6bg\n/fdx8sl00BGUAi2nOHp7pQtbZ2dUdbnA7hx0lEAvX7583LhxU6ZMGZrYz2rjwv62tratW7fKdwcS\nySkPHoyHH8ZTTwHA+PGYNw+DBuHkk3Hqqfj88ygheOwx3HADjj4an3yCSy5JJFr3sXGjNK0gYwwb\nFn8VxIcfxo03Ru5edBGefNLSoFIm2fPZNvr70xJoTQc9ahQ+/1y9caVAZ+HYhizQ3d0RBy3ulpSg\nq0tDoO1OcUQJdCAQOO+883bu3JngNFZR2A/giSee2LBhg+rZ6upq5dRY0bYmXji5EQdXV4ebbgKA\nceNwwglqZyeOmq3HLhNUVGh0FbAOed0KA8TXXsbBP1+SPZ9tI81l2zQdtNyWU7lxpjjEx+/qijho\nAMEgSkrQ2Rk1wUdQWIj29sxGGUVUXmLs2LFVVVXvvfeeGC2xGXZxgxsOgoNneDrrfDYgTQctl9kp\nHbTSVms66EyWbzoEWaB7eyMOOicnotqxDnrQIKc46I8++mjHjh179+692mBmejQLFy6cM2fON7/5\nTVhR2K/SJpVTEzPxHWzfzCHzAu2VQ5rC+WwbaTpoucxO6aBlgdZz0KoFR7IBWaALCiLJjdJS6cHY\nHHQohNJSp+Sg//u//zvZNdzkwv45c+aYU2Onh/ONpEU4/4M7dZWKFM5n27DIQQu56e9HKCRV4GW5\ng5YrWwoKpORGTw/KyiIOWpXi6O1FebmGrc4gkRTH3/72tyMKEnmzbYX98I7Ri0PmBTruWK7qyJu7\nwrR5pHA+24bVDlr+3+/PagddVIRgEP39KCmJOOjycl0H3duLQMDeGXMRBz1p0qRk60AtKeyXz9RE\ntMnbMm2wqrx1JHs6OvVPkML5bBtW1EErHxT/5+dHpDybBwmDQZSXRxy0UqBVDrqnx/ZOOBGBHj9+\n/JgxY5J6s/lfAJ9P+ydzbI8Ip+qCyThw6VKXkML5bBvppzhkBy0LtDLvAagvAN3d2ZviEK1Wxez/\nnp5IDrq/XyPFYfcXMJLiSLDjF8koU6faHUE8nHqldNP5LOaS6B3JUCjODymlWdYsswPUF4CuruxN\ncYiiOvFLsbsbFRUIBtHXh9JSjRSHcwTaKXzV7zGK2B8aIoXk+YZKymbqDkE1I9SFC3E6Dk0NVT6r\namCvQrMOWjMHja8uqFnuoOX0jjxI2NmJoUO1Uxw+n4064zyBPnxYuqHMQQ8apPYX4owUB86pJs4E\nMj9IGHePqqOd4Ax+YoCcJtZ71nj5ElmgjXPQSuQFwrMKOQctX5zkMruuLlRVaTtoMYfFJrLvj0SI\n01CZ3NhnjR208pWxAm288axCFmilgxaDhJ2dqKrSzkGLZcVtwsEC7fwSYKux5ZdBsjv18M+XjNHf\nj5yc1B208pWylCtttXLj4muVnV+uQEBqAC0LdHc3ysrQ3Y2uLlRWqhvR9PRIDtq+Mk2HCXSy5w3V\nwXaY4kif/n4EAiYItKaDVm3c7zcakPQ2gQC6ujQctJjkLY8cyvT2oqCADloHpVhnbZmdA2EO2nT6\n+hAImJDiUDpoZYpDuXHhIrOT2BSHcNAiBx1b1iIctK0Cbdna6ekzYoTdEZAE4JUyfTLpoOV+FFlI\nbIpjYAAlJejuRmcniovVrxc56NxcCrQWgwfHfw3VgXgAoaGZdNBZm4MWrfoHDYoUz4kHOzs1HLRI\ncdgq0A5OcRBXwGtk+ggNtdpBDwzA50MggMOHE1V8j6FMcRQUSJOWxYNdXRoOWh4kpIOOg8+HTZtQ\nXIyvf116RMzUFD/WNm7ErFk2tK0wha4u/PrX+Od/xkcf4bvfxSOPSMt9nXIKXn8dzmlkvGkTxPo4\nilVygCyYK5QBMuOgxbOBAFpbs7ERB4CcHIRCkkCLBWShcNAVFdJYl/zzQh4kTGzRYStwmEDLIpuT\ng+rqqKdWrkRfX0SgP/kEV1yBNWvwr/+KX/wCP/4xSkszGqpZNDbiuuvwwAP4n//Bd7+LW2+VzoZz\nz8Urr+D66zMdj97cxTVrpKXIADz5JIYMkSRg8eKMheZZTHTQ8hSMWActtpPNAi2QHbSYjax00OK2\nnOvo6UFxMQcJtfD54jvio44CkNEl+6zAOEWQSCLeXBLJTp5+euSwZ+GMYdOx1EHLGxfboUCrHLS4\nIQS6uDgqGc2JKiaQncMdxEtYmoOWN04HLVA5aJHWELpcXBw1mZB10GmhnBPl6qGqcNgFmVxXH2GH\nE9dBpzaTUOWgZYFua6NARxy0QDjooqKothucSahLIr7YM5NWBwY03JPTBFF5CfHAMXcUcR10Cr04\n/H7pFJI3zhSHQHbQynJDOuiESPyb39YWJdAOXls6OZymyzJ6Ht8zR95GrHDQsRtnikMgO+jDhyN9\njDUdtMhBi8S0TThMoGVUSi3uitI6AP392L490yFZjUvnsnOqd/qI6ot0HLQoINPMVssbFw66sBCt\nrdk7tCtmnRQWoqAA7e2SQMs56KIitYPOz4ffz37QifHaa3ZHYAFCiN0ix8QK0nfQYkhQs95D5aCL\nitDamo3LqQhECl4ItOygfT709KCwUG2WHbAmoasE2gAKHHEv6eegRXNRPQfNHLSMaGjn80kCLf+S\nEAKi6aBtxakC7fMl+itMnJFub9AVDqO9HVBkDMQZM3x4piNJJGWhPGs5YJg+mXTQWZ7ikHtFKXPQ\nYsAQoINOBoOTUumXxco9HsiEio+g+ilQW5vpMBL5LZLgmBVJkLjd7NJx0PLGhdAHAujoyF4HXVgo\nXQiVOWi55I4OOi1U2pGFi6oRTxIKoaAgrYkqoupZ02srJ6qIFEc4nL0CLeo38JWDFrcDAckeqRw0\nBVqNPL1bs4ojq3BLVt0tcToceeJfLIlM9c7Li0hw7Nvlqd7CQQNZLdDisytz0IGA1MpO5aDlFId9\n+uMwgZav/zk52SLKelUcdglflhx2pyFP/IvFxBy0cNCgQEfnoAMBqayFDjpR/H4jpVCKlwemekMR\nv+0fxPYAshN5jddYEkxxGOSgCwsRCtFBAzEOWhZo4aBVMwnl4+n32zXK5VSBhpaVyyrtyKoPS0SO\nQpPEBwn1HLQYGRPbkX/UZycqBx2b4tCcNCh39884DhbouO1GPdOLwxXE/mohJmJKikPPawv1F9vx\n+1FQQIFWO2g5xaF00DJy67uM42CBjlukQZmwhcWLdZv6k5SxwkGL+d+IkW9ZpLKQ1Bx0fr5dAu2w\nglZxAuXkoLg4ykGLx7/8EvX1GDUKJ5+MnBwMGgRAuvTdeCP+3//Dt76V+ZDT4n//Fw88AACPPoq2\nNsyejc5OnH46duzAu+/C57OhlPDwYezcKS2G8Prr+PnPEQohHMbf/ia9YMiQqNf39mLu3Ij7q6zE\n0qXSQjCig8HSpfja1zIUvHuxwkHn5Uk/zOUMtZBvudQsC1E66O5utYOWL2kq7HPQDhPo4cPxn/+J\n3/4W3/52lBDccAPeeQdPPSUttvTRRygrw+WX44IL0NyMO+7A+vWYPt19Av3mm3jxRQB4+20AeOYZ\nTJuGzZvR14cjR+D32/BF6uvD7t2SQH/wAZ58MvJUYyMqK9UhFRTgiSeiHrn8cvzud5G7c+dSoONj\nhYOWtynXeAj5Li6mQEeqoaEQaD2Y4oggDp+qiiMvL6reReRD/X6UlUVeZl9TbTMRHycvD+Xl9q+y\nqBoYGTwYgwZl6YLQVpN+FYemgxauXOWgKypMCdmVKB20/L+c4tCDDlqDBMvsiHWI9iAkA5gyUWVg\nIConJm9TdtBChu6+25SQXcnkyVJqVCnQ3/gGjjvO6F32VXE4T6Dl2gzjMUCOEDoWXj5TwBQHrULP\nQX/722kG62IqKqQfEMoUx8iRcd7FFEdyfPml3REQYiqmOGi9bapy0ARAbi5ycjSa1fl8Gu35KdBq\nYg2y8hFNr6FZwEiSRSxynA6inSNJikw6aCIoKNAYLJVLoZX5Igq0hKzCximOUEjjWW8ItKNSN3qS\noSQ2YJUBYcbDmHAYPh8dtA2IdVVUlJRIpdDKRhwU6AiJKNTAgLOEzKskItCxUJGTQhhkOujMo+eg\nRT2Yslu/fRNVnCfQSEyjlV5bQF0gbkTobwYcdCJCn1XoOWgh0HTQ2gjB/e53MXFi1OPy5NRwGH19\nGkfWjQItYnbarwG9xl0Jfr05hJsUQn8tddBiXepEhD6r0HTQskArHTTroNWcdpp6lrPxXVeT/ric\nuej1Pk3wQuKoz+J8LHXQyupdOmgVmg5abgmtctA2zYNznsylbCf1zm+SFE6z855HjN2Z0g869sHu\n7sh7OUioIhDQuPIxxWGEgTooc80e9mhO0MekDq8TAnY1YuzOohRHMBh5LwcJVaxZo/Gg7KB7eiIC\nzUHCCPzCux2Vvnv4amoKcXPQog7PGM0MSV4euroioswUh4pjjtF4sKQEHR0A0NsbSYAUFlKgFWiu\nd6V8RHMUi0JgOqkd0tSK87IWU3RTLwetdNAcJEwEhw0SOlKgNfvBK88tTQmInaBJbIFXyqQwRTc1\nDbgQaDropFBOVJEPHQU6gt6vuRUrIrfDYY3VdinQZpGmwlKgk8IsB93To/7uiBQHHXRSKAWaE1VS\nob8f4XCkN7/cNNm9upCTA78f48fbHQcAoKQk3S249w9hC2Y56O5u9XbooFNAOZPQAVUcDvuDHXUU\nursxa5aGj87Px/DhaGrCkSPo6op0CBw0CEcdhb4+vPRSonu59lr87GemxZwyO3di9WrU1uKSS/Cr\nX+GUUzAwgPp6m6PKy0M4jAMHcPbZaGwEgNpaqVBXU0dqa/H3v2PsWOnuwYNYuhSVlTh0CLW1CAax\ndi2WL8eRI8jPR28vQiFp3kRODi6/HIsXZ+6jORNTdDM3NyrdLGAOOgWUZXbKHDT7QQPAKacA0F65\nKjcXU6fi+ecxMICBAVx6qfR4YSF27MDBgxg6NKHxbgDvv29awOlw6BDa2vD882huxsGDuP9+uwMC\nAPj9CIVw8CDeeUd65PXXjRrmrl6N557Drl3S3auvxoMP4u67cd99eP99rFmD3/8+sikVe/eaGro7\nMSvFoTTLygeVZXZ00HFRDhI6wEG7MMUhSGeqN3+DGxMOm7CWilyKo7lMMpExK8Wh6aCVZXZ00InA\nbnbmoFzz29XEXTsm86R/AeOk8MQx0UEbpzjooBNBnvLDbna2QV0wEdUFJvY2j7YxpszAFg46NsWh\ndNChkHdsTQY4eBCVldJtTlRJmnTWjW9tNS8OL9LXF6WqqRl8eW1JA4EWU7ayHFNmYMfNQTutIZfz\n2b8fI0ZIt3NydFs8WoxrBTqdc9qmY62Lo7IcmmuypbARJKDs+/enuyMPYKKDNshBG0wlJ5o0NUUE\n2j5cJdCJtFJKBIdYCWc2g0aS7ahU8cd+HINNOeQPYS+ZcdCa3ZSIAW1tKC+P3LXpXHWVQJsFdcFc\nNK8xifws4B8C1jto8aBmsw6iSU4O+vsdsq6eOwU6Ly+tHLSjcFR+QxAOp5vlUC5IZqALTU2OSzdl\nHmMHrVxb2gADBy2nOOigE0QuhXYA7iy70VO0BCeqOKprx8iRyM11kEaL8RC5eLmuLo6/U10pRelo\nfj6eew4FBaiqwsGDuu/dtg2dnRg8OM2Q3U1fn3TQRJpYdbQTLF6mgzaRkhLs24eKiqgHbfqGutNB\np4mjflnn5Dir+Ekll3HVQe/EnTbN6FkiIztozXG8BIuXVYunKB/kIGGyFBdj1y4MH253HID7BLq4\n2ISNOEqgnU9ckXXUBcZ1yBKsmYVIMEMtqug066DljZMEKS7Gzp1OKOGAywRamdnUe8pFKKs4HBV/\nsouKGQTv9/NyGAc5iaGZhUi8xkMsbKhELLMib5wanSAlJdixgw7aPpwmGYFApGmq00j2yiFeLzcx\nUCXySCyyR9ZctirxBhqx7Qbl5Ibms0SPkhJs347q6qgHRQvGjOM2gTbFaTpNoI86Kmo5AkeR2gGX\n58gefXSco+20v0XmMSUHDR0HDYVM00EnSEmJRopDNMvNOG4TaJIZZN1MRKCNX0MJNsaUHDTiOWjR\n0ZskQnExvvgCw4ZFPWhTQ7usFGhHldk5EEpqJrE0B638nw46QcSiQqoctE0N7bJPoDs7MTBg/vyI\n3l60taGrC8GgxtesuxttbQgGTd5pBkg2xZHsWpGahyurkB206TloZf0Gc9CJU1KCIUPU1zObFlVx\n1UX1jDNQU4PWVo2hp6IiVFejpQVVVXE2cswx+Owz/PKXuPpqM2ObPh1//as06euee3DTTVHPnngi\nPvoI3/gGXnst8mBvL3w+E9YANJe6usjUktNPx4wZcSLMz8fcuZG706bhwgsxbpx0d/Bg9PWhpgan\nny4tdpWTE5Ghv/4VNTWoqsrqrkmyBJueg/b5IpkN5qATp6REo8bOphSHq/5m112n+1RhIY4/PiEv\nJtTB9IthczPwlVuM3bjoq7l7d9SDYrJ/IGByJGly3HFoapKO0k034bvfjfP6nBz88peRuxdcgAsu\niNwdNAg1NZg0CU8+qfHe2bPxzDPZvuSKdQ4aiswGc9CJ4ySB9laKw6am2vHJcg0iBigHCc110FAI\nNHPQiVNRgaOPVj9IgTaBxDPLGR4H08vDOmp+SubJ8o8vMDHFoemg5Y3TQSdIbS1WrVI/SIH2Pm6p\njjA3TqqwMXFTHImX2Rk4aOag04QCTRyHWdqqtx0uWogEHHTiZXYGDppVHGlCgc4oGRYFd2lQV5fd\nEWQTJk5UoYO2Dgp0utj7UzoRCY59jTN//mcsxeEwydi1a9eqVas++uijq666auPGjRnaq4kTVYwd\ntMOOtsvgRBUTcJdRJQ7rU7p8+fKpU6fW19evWLFi7dq1Gdprxhw0UxzpYNNEFW8JdOLIZRU7dmD1\nahM2aHx1FVeO2OtHIqsZZRifL1IMY0p4Pl8cIe7tdcgKQ36/v6SkxOfzFRQU+DP2p6GDdgVMcWQI\noZLyN2HnTrzyigmbVeZtY4VYSJ6q2C4cdtwsFUF/v/QRTPlK+/1RqyMrEdmP3l4cPmzCjtJm0aJF\nL7744po1axoaGs4666wM7ZUO2hVQoO2BWREDnJkit4y8vLxAIFBYWFhcXBzI2LWTDtoVuFGgN23a\ntG/fvoaGhvnz53/44YdmxURsJqleox7Cnhx0KCSlgOignYwbe3E89thjoVBo8eLFVVVVV199debO\naU1SU5NMOmjXuXVTBNpgIw67ANiTg5bJy0N3t/pBOmiH4EYH3d7eXlRUVFtbW1paWuqERZtSU0BT\ndDO1dUMcplBW4ZKFr4xz0Hv27Dn55JOnfUV3rJgquflmHHssTjpJurttG/bulW7X10s3Vq7EpEko\nLJTu1tbil7/EtGlR/379a9TWJhT9tGkauf7TT5e2X12NSZMS2g7RZORIPPccRo3CO+9kcrdpXVRv\nvPHGN954o7Gxcfv27QsWLDArpkwgK6YVrtZ1TlmF6fHHlRhnHLGJEycuWbIEwBNPPLFhwwbVs6NH\nj37zzTflu+V6I5+C3bvx8su48ELp7vPPY+hQzJ2LcBh//rP04LZt+OMfUVcn3f3GN/D3v6cevWav\nx9tvl25MnozJk1PfODn6aPz971i5UupbmSnSEugnn3xyzpw5xx577LHHHmtWQJnG9M79gG7XU2fI\nUBJkWYrDTNraUFYWuRsMSss1dHejuRkDA/D70dTkkKWjSaIUFuouu/G972HdOtN3mFaKo6OjY9++\nfcuWLdu2bZtZAWUaLn9FvmLhwoWbN28Wt2fNmpXWtkIh5ObC55OuyrJAixVkhAsLBh1aZ0n0KCrS\nFejGRit2mJaD9vl89fX1X3755SOPPLJs2bIno5uyB4PBLVu2yHdDVnhVTZqa8MEHAFBTg4kTo55q\nbJS+MHLZ8q5dOHAAmzbh+OMxZIh6Ux0dOHQIAEaMQG4u3nwT7e0YPx6jR0e9bM+eKNf86ad46SVU\nV+OYY9DYiF27JJ/e14d338WXX2LsWHz6Kd57z7SPbC6ffZbpGTR2zNGKRTYcc+bMmag6c1KjsBDd\n3QgEpLXQAOl/emeXEgjoCrQ1tfxpCfS0adMAVFVVLVq0KPbZtra2rVu3yncHrPaqVVXS9/xXv5JS\nb2ecAWVHhY4OXHstWlowaxZefBF33gkAy5bhyBH8+c/w+dDcrNbojRvxl7+gvx/XXotjjsF3voMj\nR3D++VDVq9x9N8rLcdttePJJvPsufvtb/Pa3mDEDL7+MZcvwm9+grg7z5+N//gfHHw8AJ54IcWTm\nz7fueKTIpEm49FIA+Ld/M2dY6cYbMXWq9lP19TjmGPzud3jvPYwda8K+0sPYcKSC+D7HCvT+/fja\n1zTWbyQOJxBAe7v2U3qPp0daAr1w4UL59oYNG1S/Cqurq2+55Rb57sqVK9PZV3yKiqQbeleCcFj6\nN2MGnn5a49nY3LF4PRBV/xT7slAII0bgllvwj3/g3Xc1dl1bi1tuwVNPSYvvyQusONBGySGdfz6q\nq03YoFy0EMvMmZg5Ex995JDsvLHhSILeXqm4ragIXV2oqIhKcQwahKYmHDiAoUPTj5lkFPGTKJae\nHouK8Lw4k9DAqgshSGFsSrWcYCxim8qmE8plrsSz8n5j5yM4EA+P4OmgNBxpIa9rLAQaiHLQ48bh\niy+wf7/GwnfE4ch/UBXBoEVpurQEn09acAAAFM9JREFU2sxBFRMxWDo2VruVL9bzcaqaPM2XCTkr\nKIg8snNn5LaYIyBLXrZO1csWZIGWU5ZKgR47Fk1NFGhXopeD7upCX58VPwSzrIoj1kEnZWYNLpJ6\nUqv5N3PGL3piFcYOuq4O+/dToF2JnkCLBw2sYaqkJdBiUKWhoWH9+vUXX3yxWTFZSKxAJ6WVBi8W\nX0jjtZ1S2ylxHW1t0qQ+MTyI6DK7mhq0tFCgXYleDlr8cS1IXVpYxWEDaVa6pJPiSIRY+XZyisPJ\nsTmctjZ1iqO/PzJRpagIAwM4cADDhtkZJEkBYwfd2xspVTCJdHPQZsVhDnolijJpWleznC8dtLdp\naVGnOAIByXnJk1O++MKcIhmSSQxy0LDEQXuxiiMuKfRdTEFSDd5CgfY2soOWBdrvlwaohUCXlWHP\nnqi54MQVGDtoCnQcYpcsUREOIz8fJSXS/MDYZzUf7OuLatmhWcYnpt4pqziU2ywpARCZmCAPJjgw\njSCH5LAFA91Ea2skBy1/n+U534EAhg93yCpfJDnipjjMxkMCLS+mZ5xKnjABI0eqvzZ6iGdDoajx\n2dg560L3Aej1XJ0wAQCKi3W34EAyJtBywwrPoKziUH2fZYHmLBU3kpurXarR1QWfjw46HnG/59a1\nfjbueuNApxwXN8bsEFRldmLNFHE8hUBXVzMB7SnEBFEKdBwsMmLpbFa8V9V7yBWGkQKdMt3dUpt8\n8Ys4GIwM7guBHjGCNXaeoqsLpaWOK7NzHFY7aBML7JyPG2N2GsJBd3UhEEBHB/CVQJ94IgYPtjs4\nYh7d3Rg82IoctLcEOkFKS7VXVEmn7iJWzsRbNP9mnOqdJYiJKl1dagc9ciRGjrQ1MmIqljlob6U4\nZPTaGYfD8Pk0ai2MUamzWZMPiScJhSKnnxgkFAItVh1lk35PEgxSoBNAVkO5XiLD1NSoH9Hsn+cK\n1aa7T41DhyKLt4oUh8hBy/loCrT36OqyKMXhLYGW+frXtR+3WhmHDUvanjsWCnRqyI04oEhxBAIU\naI+Qk6NRaWeZg/ZWDvqll3DOOfjwQ2kZ6d27sXMn9u5FYSHefhstLdi7F4MGAUBHB+65R50Juf9+\nlJYiEJAWSD5yBBs2YPt29PZiYCDSJEVW+WAQ99+PUAjvvYcTTgCA738fjz6K9esBoLsb99yDTz4B\nYlIucaekOwGOYqWGXGOHr2Z4KwValNwR9yL+pmLqmUwwiJEjKdCGHH00rrsO992HF17ASy/hoovw\nL/+CV1/F88+jtBSPPCK9bMYMTJyIsjIsXgwAM2fiP/4DoRCeeAJ33y29pqEBubn49FP85jfSI1u3\n4vHHkZuLDRsiL/v8c9x0k3T7V7+Sbtx9N669FgAWLpR28dZbEKue33cfmpqwbx8uvxw334wzzlAv\nmegERozApk3w+TBlSob2uGSJxmqQ7kWeRgjA58PAQFSKg7gd8XdUCbRlKQ4PCbRyhbdAAKefDr9f\ncrvKnyR+PwoKUF6OvXsBIDcXM2YAwFtvxd+Fz4eZMyMCrUTuTDZpkrSUnzyrcNQoKbbJkzF5MkTv\n7EmTMHNmEp8uY+Tk4Dvfyegex4/P6O6sRm7EISMGCSnQ3kDz7yjK7DhImChy/lRIc2zqOTeDV6ZM\n7ovYjtzKTkYIdOy0b+JG9ASaMwmTJpPDXAb74mhbVqHMQQuUOWjidjT/jqIbDwXaiDR1MMMyypEi\nr6JKcfh8zEF7Cs2/o8+HvDyW2cUjHJbW0lYtahWb4khNjpN6l/GLU+hJTVyBykGHwxRoT6H3d8zL\no4NOgFBIYyK1Qfmz/JoUJFu52WRTHMx7eBWVQOfmor1dSnG0t3NAwvXoCTRTHEmgN9XbCVCavU1H\nh1RrLygqQkuL5KBbWzlLxfUYOGimOBJFFmhXTKomXkJ1ygUCOHhQEui2Ngq062GKI0Vi51gb5BZi\nF1RWfnP0ZL2wED4fenqku8q/R+wXT2SZxTeTZC100B4jVqDF7FCmOOIgBFFvXUEVYi64EtFh3Rgx\n30ResEq5OGHs20W2sawsoS0TbyDaJSoJBNDSIuWgKdAeoLBQexkzOug46OV2jbMc1mWEmWvOQo4c\niUpAAygqikpx8GrtdkQvDiUU6KRxfntPyrcnUTbiEIiGdnl5dNAeITbFISYicZAwUWLlOLU6aPld\nZWXa72JZK1HR2orKyqhHiooi6xNSoD1ArECLOnfmoBNCVZtsindW/SyVt5nIBTP2SkDv7GE0HbRY\n74oC7Q00BZopjvgou9mpcFSKw8k12iRNYlvZiTZJAAIBdHZSoF2PQYqDAm1Egt0thIFNZwKhTGq6\nz0b4Hia2lZ1SoKFVjknchUGKgznoVFAWw+GrNs2jRkl3Y+vtCEmZ2FZ2osAOQG4ucnMp0K4nsykO\nD3UGEA76sssiqwrl5ODhh9HUFJVEFuXJ8vdEblq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      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Autocorrelation of BAC trade price changes\n",
      "\n",
      "<span>BAC data from 04-May-2012.</span>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%R\n",
      "\n",
      "#----------------------------------------------------------------------------------\n",
      "# Plot autocorrelation function of price changes\n",
      "par(mar=c(5, 4, 4, 2) + 0.1,cex=0.8)\n",
      "dp <- diff(log(p))\n",
      "ac <- acf(dp,plot=FALSE)\n",
      "plot(ac,main=\"Autocorrelation of price changes\",col=\"red\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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8qqqqXHw1jh8/vmfPnvHjx5eUlISEhLSi1WiVPyyppqbmmWeesf4ZmJiY2LNn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\n"
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### BAC signature plot\n",
      "\n",
      "<span>BAC data from 04-May-2012.</span>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%R\n",
      "\n",
      "# Compute RV as function of sampling frequency q\n",
      "nn <- 1000\n",
      "rv <- sapply(1:nn,function(q){rCov(diff(log(p),lag=q))/q})\n",
      "\n",
      "#----------------------------------------------------------------------------------\n",
      "# Signature plot\n",
      "m <- 500\n",
      "par(mfrow=c(1,2),cex=.6,mar=c(5,4,1,2)+.1)\n",
      "plot(1:m,rv[1:m],col=\"red\",type=\"l\",ylab=\"RV estimate\",xlab=\"Sampling frequency q\")\n",
      "plot(20:m,rv[20:m],col=\"red\",type=\"l\",ylab=\"RV estimate\",xlab=\"Sampling frequency q\")\n",
      "par(mfrow=c(1,1))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Rox9++GGOrUlvOrqfBWMHTYqd81HvpUuXzp4922QytfPNVVVV/fv3f/755y+99FIvH1PP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\nlBTZ5RAxoInO6Nu3b+fOnbt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      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Oomen\u2019s noise-to-signal ratio\n",
      "\n",
      "Following Oomen, we define the noise-to-signal ratio:\n",
      "\n",
      "$$\\xi=\\frac{\\mbox{Var}[\\epsilon]}{\\sigma^2}.$$ \n",
      "\n",
      "$\\xi$ may be efficiently estimated using\n",
      "\n",
      "$$Var[\\epsilon]\\sim -\\frac{1}{T}\\,\\sum^T\\,(Y_{i+1}-Y_{i})(Y_i-Y_{i-1})$$\n",
      "\n",
      "which is just the square of the half-spread in the Roll model.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### The Roll model\n",
      "\n",
      "Recall that in the Roll model, with\n",
      "\n",
      "$$p_t=m_t+\\epsilon_t,$$\n",
      "\n",
      "the\n",
      "effective half-spread is given by\n",
      "\n",
      "$$c=\\sqrt{-\\gamma_1}$$ and\n",
      "$$\\sigma^2=\\gamma_0+2\\,\\gamma_1$$\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### The conventional solution\n",
      "\n",
      "-   The conventional solution is to sample at most every five minutes or\n",
      "    so.\n",
      "\n",
      "    -   In our BAC dataset, there are 17,858 trades. Then there are 229\n",
      "        trades every 5 minutes.\n",
      "\n",
      "    -   We get an annualized volatility of around $33\\%$. Compare this\n",
      "        with $39\\%$ and $40\\%$ 1-month historical and implied\n",
      "        respectively on 04-May-2012.\n",
      "\n",
      "    -   Sampling only every 5 minutes corresponds to throwing out 99.6%\n",
      "        of the points!\n",
      "\n",
      "\n",
      "-   To quote Zhang, Mykland and A\u00eft-Sahalia, \u201cIt is difficult to accept\n",
      "    that throwing away data, especially in such quantities, can be an\n",
      "    optimal solution.\u201d\n",
      "\n",
      "\n",
      "-   From a more practical perspective, if we believe that volatility is\n",
      "    time- varying, it makes sense to try and measure it from recent data\n",
      "    over the last few minutes rather than from a whole day of trading.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Subsampling\n",
      "\n",
      "-   Denote by $Y_{jk}\\,\\,(1 \\leq j \\leq k)$, the subsample of $Y$\n",
      "    obtained by sampling every $k$ ticks from the $j$th tick on.\n",
      "\n",
      "-   There are clearly $k$ such non-overlapping subsamples in the\n",
      "    dataset, the first one beginning with the first tick, the second\n",
      "    with the second tick and so on.\n",
      "\n",
      "-   Then, for each $Y_{jk}$, we have \n",
      "\n",
      "    $$[Y_{jk},Y_{jk}]\\sim\n",
      "    \\left[X_{jk},X_{jk}\\right]+2\\,n_{jk}\\,\\mbox{Var}[\\epsilon]$$\n",
      "    \n",
      "-   Now define the average subsampled realized variance\n",
      "    $$[Y,Y]_k:=\\frac{1}{k}\\,\\sum_j^k\\,[Y_{jk},Y_{jk}]$$ and suppose\n",
      "    further that the *per tick* variance $\\sigma^2$ is constant.\n",
      "\n",
      "-   Then \n",
      "\n",
      "    $$[Y,Y]_k \\sim \\bar n_k\\,k\\,\\sigma^2+2\\,\\bar n_k\\,Var[\\epsilon]$$\n",
      "    \n",
      "    where $\\bar n_k$ is the average number of ticks\n",
      "    in each subsample.\n",
      "\n",
      "-   We obtain the estimator\n",
      "\n",
      "    $$v^{Realized}_k= \\frac{[Y,Y]_k}{\\bar n_k\\,k}$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### The ZMA estimator\n",
      "\n",
      "With our earlier assumption of iid noise, we can eliminate bias for each\n",
      "$k$ by forming\n",
      "\n",
      "\\begin{eqnarray*}\n",
      "[Y,Y]_k-\\frac{\\bar n_k}{n}\\,[Y,Y] \\sim \\bar n_k\\,k\\,\\sigma^2-\\bar\n",
      "n_k\\,\\sigma^2\\end{eqnarray*}\n",
      "\n",
      "Thus we obtain the Zhang-Mykland-A\u00eft-Sahalia (ZMA) bias-corrected\n",
      "estimator of $\\sigma^2$:\n",
      "\n",
      "$$v^{ZMA}_k:=\\frac{1}{\\bar n_k\\,(k-1)} \\, \\left\\{[Y,Y]_k-\\frac{\\bar\n",
      "n_k}{n}[Y,Y]\\right\\}$$\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### The Zhou estimator\n",
      "\n",
      "Define\n",
      "\n",
      "\\begin{eqnarray*}\n",
      "[Y,Y]^Z: = \\sum_i^T\\,(Y_i-Y_{i-1})^2\n",
      "+ \\sum_i^T\\,(Y_i-Y_{i-1})(Y_{i-1}-Y_{i-2}) \n",
      "+ \\sum_i^T\\,(Y_i-Y_{i-1})(Y_{i+1}-Y_{i})\n",
      "\\end{eqnarray*}\n",
      "\n",
      "It\u2019s easy to check that under the assumption of serially uncorrelated\n",
      "noise independent of returns,\n",
      "\n",
      "$$\\mathbb{E}\\left[[Y,Y]^Z\\right]=\\langle X\\rangle_T$$\n",
      "\n",
      "It follows that\n",
      "\n",
      "$$[Y_{jk},Y_{jk}]^Z \\sim (n_{jk}-2)\\,\\sigma^2$$\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
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       "slide_type": "slide"
      }
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     "source": [
      "### Subsampled Zhou estimator\n",
      "\n",
      "As suggested by Zhou himself, we may compute his estimator from\n",
      "subsamples of the data obtaining\n",
      "\n",
      "$$v^{Zhou}_k:=\\frac{1}{\\bar n_k-2}\\,[Y,Y]^Z_k$$\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Notation\n",
      "\n",
      "-   Given an observed price series $\\{p_i\\}_{i=0}^T$, let\n",
      "\n",
      "    <a name=\"Eqn:gamma\"></a>(1)\n",
      "    $$\\gamma_{h,q}(k) = \\sum_{i=1}^{m} (p_{iq+h}-p_{(i-1)q+h})(p_{(i+k)q+h}-p_{(i-1+k)q+h}),$$\n",
      "\n",
      "    where $m = \\lfloor (T-h+1)/q\\rfloor-k$.\n",
      "\n",
      "    -   Thus $q$ is the sub-sampling frequency, $h$ is the index of a\n",
      "        given subsample and $k$ is a time-offset.\n",
      "\n",
      "-   Many of the realized variance measures are derived in a setting\n",
      "    where observed prices are modeled as $p_{t_i} = \\int_0^{t_i}\n",
      "    \\sigma_u\\,dW_u+\\omega\\,\\varepsilon_{t_i}$ with $W\\perp \\varepsilon$.\n",
      "\n",
      "    -   Their implementation typically requires measurements of the\n",
      "        latent integrated variance $IV=\\int_0^1\\sigma_u^2du$, integrated\n",
      "        quarticity $IQ=\\int_0^1\\sigma_u^4du$, and noise variance\n",
      "        $\\omega^2$.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Mean-squared error\n",
      "\n",
      "-   Mean-squared error (MSE) of an estimator $\\theta$ of some quantity\n",
      "    $\\theta$ is defined by:\n",
      "    \n",
      "    $$\\text{MSE}[\\hat \\theta]=\\mathbb{E} [(\\hat \\theta- \\theta)^2]$$\n",
      "\n",
      "\n",
      "-   Bias is given by $\\mathbb{E} [\\hat \\theta -\\theta]$.\n",
      "\n",
      "\n",
      "-   Variance of the estimator is given by\n",
      "\n",
      "    $$\\mathbb{E} [(\\hat \\theta-\\mathbb{E} [\\hat \\theta])^2]$$\n",
      "\n",
      "\n",
      "-   Then\n",
      "\n",
      "    \\begin{eqnarray*}\n",
      "    \\text{MSE}&=&\\mathbb{E} [(\\hat \\theta- \\mathbb{E} [\\hat \\theta]+\\mathbb{E} [\\hat \\theta]-\\theta)^2]\\\\\n",
      "    &=&\\mathbb{E} [(\\hat \\theta- \\mathbb{E} [\\hat \\theta])^2]+(\\mathbb{E} [\\hat \\theta-\\theta])^2\\\\\n",
      "    &=&\\text{Variance}+\\text{Bias}^2\\end{eqnarray*}\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Estimator consistency and efficiency\n",
      "\n",
      "-   An estimator $\\hat \\theta$ is <span>*consistent*</span> if it\n",
      "    converges in probability to the true value $\\theta$.\n",
      "\n",
      "-   Estimator $A$ is more <span>*efficient*</span> than estimator B if\n",
      "    $\\mathbb{E} [(A-\\theta)^2]<\\mathbb{E} [(B-\\theta)^2]$.\n",
      "\n",
      "-   An <span>*efficient*</span> estimator is the most efficient one.\n",
      "\n",
      "\n",
      "-   An article in econometrics describing a new estimator typically\n",
      "    proves that this new estimator is asymptotically consistent and\n",
      "    efficient in the limit that the number of samples $n \\to \\infty$.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Optimal sampling frequency\n",
      "\n",
      "-   The optimal sampling frequency is such that MSE is minimized.\n",
      "\n",
      "    -   As sampling frequency increases, the variance of the estimate of\n",
      "        realized variance decreases but its bias typically increases.\n",
      "\n",
      "    -   This optimal sampling frequency may been computed theoretically\n",
      "        for various estimators of realized variance under idealized\n",
      "        assumptions.\n",
      "\n",
      "-   Analogously, for the estimators that we now describe, there is\n",
      "    typically a parameter (such as the subsampling frequency $q$) that\n",
      "    theoretically minimizes MSE under given assumptions for the\n",
      "    underlying and noise processes.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Zhou and ZMA estimators in updated notation\n",
      "\n",
      "With our updated notation:\n",
      "\n",
      "-   The Zhou estimator becomes\n",
      "\n",
      "    <a name=\"Eqn:ZHOU\"></a>(2)\n",
      "    $$ZHOU = \\frac{T}{T-q+1}\\frac{1}{q}\\sum_{h = 0}^{q-1} (\\gamma_{h,q}(0)+2\\gamma_{h,q}(1)),$$\n",
      "\n",
      "-   The ZMA (or two-scale RV) estimator becomes:\n",
      "\n",
      "    <a name=\"Eqn:TSRV\"></a>(3)\n",
      "    $$TSRV=\\left( 1-\\overline{T}/T\\right) ^{-1}\\left( \\frac{1}{q}\\sum_{h = 0}^{q-1}\\gamma\n",
      "    _{h,q}(0)-\\frac{\\overline{T}}{T}\\gamma_{0,1}(0)\\right),$$\n",
      "\n",
      "    where $\\overline{T} =(T-q+1)/q$\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Multiscale RV\n",
      "\n",
      "Zhang\u2019s multiscale RV estimator is given by\n",
      "\n",
      "<a name=\"Eqn:MSRV\"></a>(4)\n",
      "$$MSRV = \\sum_{j = 1}^{q} \\frac{a_j}{j} \\sum_{h = 0}^{j-1}\\gamma_{h,j}(0),$$\n",
      "\n",
      "where\n",
      "\n",
      "$$a_j^\\ast = (1-1/q^2)^{-1}\\left(\\frac{j}{q^2}h(j/q)-\\frac{j}{2q^3}h^\\prime(j/q)\\right)$$\n",
      "\n",
      "and $$h(x) =12\\,(x-1/2).$$\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Realized Kernel estimator of Barndorff-Nielsen, Hansen, Lunde and Shephard\n",
      "\n",
      "<a name=\"Eqn:KRV\"></a>(5)\n",
      "$$KRV = \\gamma_{0,1}(0)+2\\sum_{s = 1}^q \\kappa\\left(\\frac{s-1}{q}\\right)\\gamma_{0,1}(s).$$\n",
      "\n",
      "Choices for the kernel $\\kappa(x)$ include:\n",
      "\n",
      "-   Modified Tukey-Hanning kernel TH$_{2}$: $\\kappa\\left( x\\right)\n",
      "    =\\sin ^{2}\\{\\frac{\\pi }{2}\\left( 1-x\\right) ^2\\}$.\n",
      "\n",
      "-   Cubic kernel: $\\kappa\\left( x\\right)\n",
      "    =1-3x^{2}+2x^{3}$.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Large\u2019s alternation estimator\n",
      "\n",
      "\n",
      "<a name=\"Eqn:ALT\"></a>(6)\n",
      "$$ALT = \\frac{C}{R}\\,\\gamma_{0,1}(0),$$\n",
      "\n",
      "where $R$ ($C$) are the number of reversals (continuations) in the\n",
      "sample.\n",
      "\n",
      "-   Note that $C =\n",
      "    \\sum_{i=2}^{T} (I^p_i I^p_{i-1}+I^n_{i-1} I^n_i)$ and\n",
      "    $R = \\sum_{i=2}^{T}( I^p_i\n",
      "    I^n_{i-1}+I^p_{i-1} I^n_i)$ where $I^p_i = I(r_i>0)$ and\n",
      "    $I^n_i = I(r_i<0)$.\n",
      "\n",
      "-   If there are zero returns in the sample then these are first\n",
      "    removed, *i.e.*, the estimator is implemented using tick\n",
      "    data.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Large\u2019s estimator\n",
      "\n",
      "-   Suppose that $$\\Delta p_t=p_t-p_{t-1}=\\pm k$$ for some fixed number\n",
      "    $k$. That is, the trade price jumps up and down by a constant (1\n",
      "    cent say).\n",
      "\n",
      "\n",
      "-   Recall that \n",
      "\n",
      "    $$\\gamma_{0,1}(0)=\\sum_t\\,\\Delta p_t^2=N\\,k^2$$ \n",
      "    \n",
      "    which is just $RV$. So \n",
      "    \n",
      "    $$ALT = \\frac{C}{R}\\,RV$$\n",
      "\n",
      "    -   If the efficient price is constant, then $C=0$ and $ALT=0$.\n",
      "\n",
      "    -   If there is no microstructure noise, $C=R$ for a random walk and\n",
      "        $ALT=RV$.\n",
      "\n",
      "\n",
      "-   This estimator only works for large tick markets where the price\n",
      "    moves by only one tick whenever it moves.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Maximum Likelihood estimator of A\u00eft-Sahalia, Mykland and Zhang\n",
      "\n",
      "<a name=\"Eqn:MLRV\"></a>(7)\n",
      "$$MLRV =  M\\,\\widehat{\\delta}^2\\,(1+\\widehat{\\eta})^2,$$\n",
      "\n",
      "where $(\\widehat{\\eta},\\widehat{\\delta^2})$ are the maximum likelihood\n",
      "estimates of an MA(1) model for observed returns, <span>*i.e.*</span>\n",
      "$r_i = \\varepsilon_i+\\eta \\varepsilon_{i-1}$ where the $\\varepsilon_i$\u2019s\n",
      "are serially uncorrelated with mean zero and variance $\\delta^2$.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Practical implementation\n",
      "\n",
      "-   Ideally, we should be able to update a RV estimator in real time as\n",
      "    ticks come in.\n",
      "\n",
      "-   From this perspective, estimators where we can just add the most\n",
      "    recent observation and drop the oldest observation are preferred.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Convergence of TSRV, MSRV and kernel estimators\n",
      "\n",
      "-   The TSRV two-timescale RV measure yields a consistent estimator of\n",
      "    IV that converges at rate $M^{-1/6}$.\n",
      "\n",
      "\n",
      "-   The rate of the TSRV estimator can be improved to $M^{-1/4}$ \u2013 the\n",
      "    fastest attainable in this setting \u2013 by using multiple time scales\n",
      "    as in the MSRV.\n",
      "\n",
      "\n",
      "-   The realized kernels provide an equally efficient alternative to the\n",
      "    subsampling estimators with rates of convergence of $M^{-1/6}$ or\n",
      "    $M^{-1/4}$ depending on the choice of kernel (the TH$_{2}$ and cubic\n",
      "    kernels converge at the fastest rate).\n",
      "    \n",
      "\n",
      "-   Finally, both the ALT and MLRV estimators are also consistent and\n",
      "    converge at rate $M^{-1/4}$, albeit under more restrictive (semi-)\n",
      "    parametric assumptions.\n",
      "\n",
      "\n",
      "-   An important feature of the non-parametric RV measures TSRV, MSRV,\n",
      "    and KRV is that they allow for stochastic volatility, leverage and\n",
      "    can be made robust to dependent noise.\n",
      "\n",
      "    -   ZHOU is biased with dependent noise\n",
      "\n",
      "    -   ALT rules out leverage effects and requires uncorrelated noise\n",
      "\n",
      "    -   Although MLRV can be modified to take account of dependent noise\n",
      "        it does not allow for stochastic volatility."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Using the mid-quote\n",
      "\n",
      "-   What if quote data is available?\n",
      "\n",
      "\n",
      "-   According to practitioners, using mid-quotes eliminates bid-ask\n",
      "    bounce.\n",
      "\n",
      "    -   If we use mid-quotes, do we get the same volatility estimate?\n",
      "\n",
      "    -   When should we sample the mid-quotes? Every quote change? Every\n",
      "        second?\n",
      "\n",
      "\n",
      "-   According to Bouchaud, Gefen, Potters and Wyart(2004) and also Bandi \n",
      "    and Russell (2006), it's best to sample the mid-quote just before each \n",
      "    trade.\n",
      "\n",
      "\n",
      "-   Let\u2019s see how the autocorrelation plot and noise plot look with\n",
      "    mid-quotes.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### BAC trades and quotes\n",
      "\n",
      "<span>BAC data from 04-May-2012. Mid-quotes in green, trade prices in\n",
      "red.</span>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%R\n",
      "\n",
      "#----------------------------------------------------------------------------------\n",
      "# Compare trades and mid-quotes\n",
      "\n",
      "bid <- as.numeric(tqBAC$BID)\n",
      "ask <- as.numeric(tqBAC$OFR)\n",
      "mid <- (bid+ask)/2\n",
      "\n",
      "par(mfrow=c(1,2),mar=c(5,4,1,2)+.1,cex=.6)\n",
      "plot(mid,col=\"dark green\",type=\"l\",ylab=\"Price\",xlab=\"Trade #\")\n",
      "points(p,col=\"red\",type=\"p\",pch=\".\")\n",
      "plot(100:200,mid[100:200],col=\"dark green\",type=\"l\",ylab=\"Price\",xlab=\"Trade #\")\n",
      "points(100:200,p[100:200],col=\"red\",type=\"p\",pch=20)\n",
      "par(mfrow=c(1,1))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Cfiv73r5/C4JA4oVQF61GS29CE3wp6M0Dt5zv+HfVMlUBnL95vmphFz4kfh2c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\nsHkzVq6UV9AhpKC5hhS0mxiXjVg7YgECy8S/KtSUv8YOYql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      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Autocorrelation of BAC mid-quote changes\n",
      "\n",
      "<span>BAC data from 04-May-2012.</span>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%R\n",
      "\n",
      "#----------------------------------------------------------------------------------\n",
      "# Plot autocorrelation function of mid-quote changes\n",
      "dm <- diff(log(mid))\n",
      "acm <- acf(dm,plot=F)\n",
      "plot(acm,main=\"Autocorrelation of mid-quote changes\",col=\"red\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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aACRFQAOApAhoAJAUAQ0AkiKgAUBSBDQASIqABgBJEdAAICkCGgAkRUADgKQI\naACQFAENAJIioAFAUgQ0AEiKgL7VkiWivNzWRQCAEAR0S1euiPp6WxcBAEIQ0AAgLQIaACRFQAOA\npAhoAJAUAQ0AktI1oC0WS01NjcVi0XNQALBTegR0fX19enp6WFiYm5ubl5eXq6traGjo0qVLTSaT\nDqMDcGh5eeL9921dxD3SI6BTU1Pz8/MzMzNLS0sbGhrKysqys7MLCgrS0tJ0GB0arl0TGRm2LqIT\nURRRUWHrItCKa9fE1au2LuJeKR3Py8vr8uXLLRqvX7/u5+fX3k0tWrQoIOBQfLwSH6988YWiKMrh\nw4r6VKPF+6D2Mrt3H37mr9pr+R+Pf+zmnbds7y2P3Yz3P669zJKtytGjtq/QvlrOnDk8balE9fzA\nlpF18eEXJKrnB7Zs2nT4hdUdN1bfvsePHj2jdAw9Anro0KHvvvtui8acnJyoqKj2bmrRokWzZs26\n26Xj4rTbd+9W0tO1u+bOVYqL21uVjZ08qRw92r5ViouVuXO1u9LTld27f3BNd+HKFT1Gub+SkhSL\nRaP99Gnl6ad1r6bDfP218l//pd2Vk6NUV2t3vflmx1X0g2zapLz2Wsdt/le/+tWpU6c6aON6XOLI\nysp68cUXIyIiZs2aNW/evFmzZg0ZMuTZZ5/NzMzUYfTO49gx7fZDh8RXX2l3vfSSMJs7rqJ7d/Om\nmDfP1kW0X2WlrSu4r37/+3av8vHHoqxMu2vDhh9Yzg/15pv3bVPXr4s33rhvW/sB9Ajo6Ojo4uLi\nlStXjhs3LjQ0dNy4cStWrDh//vywYcN0GL3z+PWv273K/v2SBrTFIriZ5/5qamr3Kvn5HVCH7Wzc\neN82df16qyc9+nLRaRgXl/j4+OYtly5dOnz4cEJCQmurFBcX79u3r0VjUVFReXn5+vXr72rU8nKh\nueQ334hvv9XuOn9ebN8ufH01usrKRK9eGu21tcLFRbi7t2MVs1lcu6Y9SmWl6NFDODtrdLW2O3v3\niro67a4rV8TGjcLltqNcWSnOn9de5dtvvx/rdq3tTkODuHFD9OihXbOfnzAYWrabTKKkRLuAe5jP\npiZRUyN8fDS6KiuFt7dw0joRaW1rN24IRRHdu2uvsn69xu5cuSLOntXenaoq4eEhunRpRwEmk6iv\nF15eGl3l5aJnT432mzfFO++IhQs1uqqrRcDlEJQAAAtfSURBVNeuws1Ne2uaNZ88KU6c0O46e1bs\n2CH69GnH1q5fF0K0Op+aM2CxiKoq7TfI1avC01PjJd1GAQcOfH/g7r6A6mpx4YL2KrcpLi42d9hp\nkE4Bfbv8/Pzk5OS6urrWFigtLT1z5kyLxi5duri6ut7eri0mRmguabGIgADtrogIUVkpqqtbtpvN\n4r33xK9+pbHKvn3C3V0MGaLRlZkpUlI02ktLxYEDYsoUja6tW0V0tPD31+hqbXfc3YWLi3bXI4+I\n8+c1AqWpSUREaK8SECAsFu2u1nbn1Clx4YIYM0aja80akZAgjMaW7YoioqO1R/n6a+HlJSIi2lHA\npUvi6FExaZJG1+bN4tFHtUOtta0dPiyamsTw4RpdMTHi7FmN9sZGMWiQ9u588okYOFAEB7ejgO++\nE1euiFGjNLpWrxbTp4tu3Vq2WyytHtAvvxQ9e4rwcI2u1l5RDQ0iKEi7a+BAUV0tbtxox9auXhUG\ng/D21uhqbQbKy8WePSIxUaPr44/FkCGib992FNCli/Dz0+7KzhazZ2v862U2iyFDtFe5TW1tbWNj\n490seS866Np2B1m3bt1bb71lg4FNJmXCBO2ut99W1q7V7po0Sbu9vl45fVq7a+FC5ciR9teni9Y+\ndP3kE+XVV7W7vvtOMZvbN8rKlcrGje0r4MAB5YUXtLsWLFC+/Va7a9w47fZ//EPJzGy7xnZ48UVl\nzx7trtZ2Z9s25c9/1u6aPl2prGxfAcuWKVu3tm8V3Ywdq91eWKjMn6/dtWSJsn//fSvgwAHtT33b\nw+4/JLRyuG8Sbt+u3e7mJvr10+7q0kX7z2GZde8uPD21u8LCtC8vyOCjj2xdgcNbvdrGBURHa/x9\nKRO+SSiZZcvE4MG2LqIVL72k3R4TI5577r6N0ru39sXH+07zmjX0pHkpTwhhNIr+/fUtRVYddGbe\n3Ny5c8ePH5+Xl1dRUdHQ0FBZWfnVV18lJiY+3f5bR9etWxcZec4Gt7t/2qBMmKC9TPiF+AdLpbgh\nv9O3eB/UXuadE8oLL2ivFXE5fpSpfWOtL4r/cc19qznmenxco/YycXHaaz1UHh96TnvLo/6gVFa2\nr56w81/8ea8sR/CHtzz1Wvwj1RLV0wm+qHIfv0los2vQFouyZYt218mT9vfdFjvV2kXb48eVl1/W\nt5T7QZ9r0A0N7f4YQGZ79igVFbYu4hZ2fw06JCRk586dLRp37doVFBSkw+j3h8Egpk7V7goNFXa0\nI51SRESrl19kFhurxyhdusj7McA9ePRRna6AyUGP2+yysrKmTp2akZERGRlpNBpra2sLCwsrKytz\nc3N1GB2dx6xZtq7gvnr5Ze328HCuj0OlR0Cr3yTMy8s7c+ZMVVWVt7d3SkpKXFyci+bd5kBr5s+3\ndQW66N+fj8igstk3CQG0m6ur9pfo0ElxsAH7sWaNrSuArjrRpwcA0LkQ0AAgKQIaACRFQAOApAho\nAJAUAQ0AkiKgAUBSBDQASIqABgBJEdAAICkCGgAkZVAUxdY1tMMnn3zy7LPPerb2C3i3Kiws7Oh6\nJNfY2NjF7n7h8L5qampydnY2yP27cx3KbDYbDAanzvRfQreTxWLx9fXt06dPB22/pqYmLy8vICCg\nIzZuZwHdLmPGjNm9e7etq7CliRMnbt261dXV1daF2ExaWtpzzz03aNAgWxdiMxkZGeHh4QkJCbYu\nxGY2b9587ty5RYsW2bqQe+G4/64CgOQIaACQFAENAJIioAFAUgQ0AEiqMwe0g99hJoRw8DvMhBBO\nTk7Ozs62rsKWnJ2dmQH7nYHOfJudyWRyc3OzdRW2xAwwA42Njc7Ozo58H7TZbLZYLHZ6utaZAxoA\n7Jrj/rsKAJIjoAFAUgQ0AEiKgAYASRHQACApAhoAJEVAA4CkOmdA/+tf/3r44Ye9vb2Tk5NNJpOt\ny9HbpEmTTpw4YX3qULOxZcuW8PDw7t27jxkzxvqLDQ41A++9996PfvQjo9GYmJhYUVGhNjrUDKhO\nnDjh4eFhfWqvM6B0Oo2NjQEBAX//+98vXboUHx//xz/+0dYV6efTTz+dN2+eEKKwsFBtcajZKCkp\nMRqN27Ztq66u/sMf/hAZGak42AycOnXKy8tr//79V69enThxYlpamuJgM6BqamoaOXKks7Oz+tR+\nZ6ATBvSnn346aNAg9XFeXl5oaKht69HTsmXL/uM//qNbt27WgHao2di0adNjjz2mPjaZTAaD4erV\nqw41A9nZ2T/5yU/Ux2vXrlVnw6FmQPXaa6/NnDnTGtD2OwMutj6Dv/9Onz794IMPqo8jIyPPnj1r\nsVgc5P8ieOGFF4QQmzdvtrY41GyMGzcuJiZGfbx3796QkJAePXo41Az84he/+PnPf64oSnV19aef\nfjpy5EjhYK8BIcTp06f/93//9+OPP87JybG22OkM2EGJ7VVVVWU0GtXHnp6eTU1NdXV1ti3Jhhxq\nNoxGY69evRRF2bJly5w5c1auXGkwGBxqBtT/ve+jjz7y9vbevn27+g+2Q82AxWJJSUlZvnx585+W\ntt8Z6IQB7e3tXVtbqz6ura11dnZu/lmBo3G02aisrJwxY8Yrr7yyefPmKVOmCMebASHErFmzKioq\nfv3rX0+YMEE42AxkZWUFBARMnjy5eaP9zkAnDOh+/fpZP74/ceJESEiIXfwt00EcajZMJtOECRMG\nDRq0b9++6OhotdGhZiArK+v9998XQvj6+j7zzDNHjx41mUwONQOff/55bm6un59fWFiY2Wz28/Pb\nu3ev/c6AfVTZLnFxcZWVlVu2bLl58+by5ct/8Ytf2LoiW3Ko2di8ebPZbE5JSblw4cK5c+fOnTtn\nNpsdagb8/PxeffXVkydP1tXVrVixYtiwYW5ubg41A2+99da333575MiRvLw8JyenI0eOREVF2fEM\n2PpTyg6xf//+IUOG+Pr6Jicn19fX27ocvQUGBlrv4lAcaTZ+85vftHh5l5eXK440AxaL5cUXX+zd\nu7eHh8eECRNOnjyptjvODFiVl5db7+JQ7HYG+A/7AUBSnfASBwB0DgQ0AEiKgAYASRHQACApAhoA\nJEVAA4CkCGgAkBQBDQCSIqABQFIENABIioAGAEkR0AAgKQIaACRFQAOApAhoAJAUAQ0AkiKgAUBS\nBDQASIqABgBJEdDotOrq6gwGQ0VFha0LAe4RAQ0AkiKg4XD++te/9u3b193dfeTIkUVFRWrjqlWr\nQkJCQkJCsrOzQ0JCbFog8D0CGo6lrKxs0aJFH3zwwYULFwYOHLh8+XIhREFBwW9/+9v169d/9dVX\nmZmZtq4R+B4BDcdiNBpPnDgxevRoo9EYEBBw7do1IcS6deuSk5OHDx8eGBi4ZMkSW9cIfM/F1gUA\nunJzc1u7dm1ubq6zs7Obm1vPnj2FEJcuXRo+fLi6QHBwsE0LBP4/zqDhWDZs2LBx48YtW7bs2bMn\nOTlZbezTp09xcbH6+OLFizYrDrgVZ9Do5Kqrq11cvn+du7u7l5aWurq6GgyG/Pz8lStX+vj4mM3m\npKSkyZMn//SnPw0MDFyxYoVtCwasOINGJzdgwADvf/vHP/4xd+5cd3f34ODg559//pVXXjl58uSa\nNWuio6PT09MTEhJGjRqVmppqNBptXTUgBGfQ6MQ8PDwURbm9/bPPPrM+PnfunBDi/PnzsbGxJSUl\nQogDBw706tVLrxqBtnAGDYjz58/PmDHj8uXLDQ0Ny5Yte/zxx21dESAEAQ0IIUaNGjV37tyoqKjg\n4GBXV9cFCxbYuiJACCEMmn8DAgBsjjNoAJAUAQ0AkiKgAUBSBDQASIqABgBJEdAAICkCGgAkRUAD\ngKQIaACQFAENAJIioAFAUgQ0AEiKgAYASRHQACApAhoAJPX/AMryNtBNbktKAAAAAElFTkSuQmCC\n"
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "-   We see higher order autocorrelations in the mid-quote data."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### BAC signature plot using mid-prices\n",
      "\n",
      "<span>BAC data from 04-May-2012.</span>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%R\n",
      "\n",
      "# Compute RV as function of sampling frequency q\n",
      "nn <- 10000\n",
      "rvm <- sapply(1:nn,function(q){rCov(diff(log(mid),lag=q))/q})\n",
      "\n",
      "#----------------------------------------------------------------------------------\n",
      "# Signature plot with mid-prices\n",
      "par(mfrow=c(1,2),cex=.6,mar=c(5,4,1,2)+.1)\n",
      "plot(1:nn,rvm,col=\"dark green\",type=\"l\",ylab=\"RV estimate\",xlab=\"Sampling frequency q\")\n",
      "plot(1:100,rvm[1:100],col=\"dark green\",type=\"l\",ylab=\"RV estimate\",xlab=\"Sampling frequency q\")\n",
      "par(mfrow=c(1,1))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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i4uISEhLu37/v4eERGxtrdDsWcidhZZO6TPp+//cLIxbSDiI0vCnQ1Z96lVeQ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\nAICj/g9ehdNeKdAAZgAAAABJRU5ErkJggg==\n"
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### ZI simulation\n",
      "\n",
      "-   Given a set of tick data, what was the realized variance?\n",
      "\n",
      "\n",
      "-   If we don\u2019t know what the true value was, how can we test the\n",
      "    performance of different estimators?\n",
      "\n",
      "    -   One way is to simulate from a model which could be stochastic\n",
      "        volatility or jump-diffusion for example.\n",
      "\n",
      "    -   A noise process is typically added to the efficient price\n",
      "        process.\n",
      "\n",
      "\n",
      "-   [Gatheral and Oomen]<sup id=\"cite_ref-GatheralOomen\" class=\"reference\"><a href=\"#cite_note-GatheralOomen\"><span>[</span>3<span>]</span></a></sup> generated simulations from the Smith Farmer\n",
      "    zero-intelligence (ZI) model. This has the benefit that\n",
      "    microstructure noise is in some sense much more realistically\n",
      "    modeled.\n",
      "\n",
      "\n",
      "-   We tested many different estimators and came up with a ranking.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Gatheral Oomen results for mid-quotes\n",
      "\n",
      "<h2><img src=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2015/03/ziResults12.png\" align = \"center\" width=750></h2>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Gatheral Oomen results for mid-quotes\n",
      "\n",
      "<h2><img src=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2015/03/ziResults13.png\" align = \"center\" width=750></h2>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Conclusion of the ZI simulation: Which estimators are best?\n",
      "\n",
      "-   Sample prices at the highest available frequency and then measure\n",
      "    realized variance using one of:\n",
      "\n",
      "    -   The Two-Scale ZMA estimator with a subsampling frequency of 5,\n",
      "\n",
      "    -   The Multi-Scale RV of Zhang with 5 subsamples,\n",
      "\n",
      "    -   The Realized Kernel of BNS with a bandwidth of 5.\n",
      "\n",
      "\n",
      "-   The performance of these estimators is largely equivalent and their\n",
      "    implementation equally straightforward, so which particular one to\n",
      "    use would be a matter of taste.\n",
      "\n",
      "\n",
      "-   Relative to the widely used sum of sparsely sampled returns\n",
      "    following the \u201c5-minute\u201d rule prescribed in earlier literature, the\n",
      "    efficiency gain achieved with these is likely to be substantial.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Conclusion of the ZI simulation: What series to sample?\n",
      "\n",
      "-   In terms of data sampling, use mid-quotes.\n",
      "\n",
      "\n",
      "-   When sampled immediately prior to a trade, we ensure the same number\n",
      "    of observations as for the trade data but with a heavily reduced\n",
      "    level of microstructure noise.\n",
      "\n",
      "\n",
      "-   The micro price is also preferred over the trade data but, despite\n",
      "    some seemingly appealing features, does not seem to improve over\n",
      "    mid-quote data. At least in the ZI simulation, it appears that\n",
      "    micro-price had higher order autocorrelations.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Conclusion of the ZI simulation: What bandwidth to use?\n",
      "\n",
      "-   The rule for choosing $q$ should be to let $q$ grow with the level\n",
      "    of noise so that with little noise we compute something that is\n",
      "    close to RV and with high levels of noise we effectively reduce the\n",
      "    sampling frequency so as to mitigate its impact.\n",
      "\n",
      "\n",
      "-   The optimal bandwidth $q^\\ast$ can be computed in closed-form for\n",
      "    realized kernels, TSRV and MSRV. It is of the form:\n",
      "    \n",
      "    $$q^{\\ast }=A\\,\\sqrt{\\xi}.$$\n",
      "    \n",
      "    for some estimator-dependent constant\n",
      "    $A$ where $\\xi$ is the Oomen noise-to-signal ratio.\n",
      "\n",
      "\n",
      "-   Alternatively, $q=5$ seems to work well nearly all of the time.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Trades vs mid-quotes\n",
      "\n",
      "-   [Barndorff-Nielsen et al.]<sup id=\"cite_ref-BNS\" class=\"reference\"><a href=\"#cite_note-BNS\"><span>[</span>1<span>]</span></a></sup> point out that one way to assess the performance of an\n",
      "    estimator on real data is to see if it gives the same result on\n",
      "    trades and quotes.\n",
      "\n",
      "    -   They find that results depend on the data-cleaning protocol\n",
      "        followed.\n",
      "\n",
      "    -   It\u2019s better to take data from each exchange separately and\n",
      "        average the results rather than supposing that all data comes\n",
      "        from the same source.\n",
      "\n",
      "        -   This might be less true now post Reg. NMS (Regulation National Market System).\n",
      "\n",
      "    -   The realized kernel estimators do fine, giving similar results\n",
      "        for trades and quotes.\n",
      "\n",
      "    -   There are some days with lengthy strong trends which are not\n",
      "        compatible with standard models of microstructure noise.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### The Uncertainty Zone (UZ) model of Robert and Rosenbaum\n",
      "\n",
      "-   In a world with no microstructure noise but with prices quantized in\n",
      "    ticks, participants would trade when the efficient price crosses the\n",
      "    tick grid.\n",
      "\n",
      "\n",
      "-   The idea of the model of [Robert and Rosenbaum]<sup id=\"cite_ref-RobertRosenbaum\" class=\"reference\"><a href=\"#cite_note-RobertRosenbaum\"><span>[</span>6<span>]</span></a></sup> is that participants trade\n",
      "    only when the efficient price is sufficiently far from the last\n",
      "    traded price.\n",
      "\n",
      "    -   Only transactions with price changes are modeled.\n",
      "\n",
      "\n",
      "-   Thus, the model establishes a band of width $\\eta$ around each\n",
      "    mid-tick on the grid and price changes can occur only when the\n",
      "    efficient price exits this band.\n",
      "\n",
      "\n",
      "-   From [Figure 1](#fig:RobertRosenbaum), if the efficient price is\n",
      "    $X_{\\tau_i}$, it must be given by\n",
      "\n",
      "    <a name=\"eq:efficientPrice\"></a>(8)\n",
      "    $$X_{\\tau_i}=p_{t_i}-\\alpha\\,\\left(\\frac{1}{2}-\\eta\\right)\\,\\mbox{sign} (p_{t_i}-p_{t_{i-1}})$$\n",
      "\n",
      "    where the transaction time $t_i > \\tau_i$. $\\alpha$ is the tick size. \n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Schematic of the price process\n",
      "<a name=\"fig:RobertRosenbaum\"></a>\n",
      "\n",
      "<h2><img src=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2015/03/RobertRosenbaum.png\" align = \"center\" width=550></h2>\n",
      "Figure 1: Tick grid in green; mid-prices in red; uncertainty zones in blue.  For a transaction to occur at the grid point $L_0$ at time $t_i$, the efficient price $X_t$ must breach the uncertainty zone above or below.  Conditional on observing $L_0$ at time $t_i$, the efficient price must have been $L_0 - (1/2-\\eta)\\,\\text{sign}(p_{t_{i}}-p_{t_{i-1}})$ at time $\\tau_i < t_i$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Schematic of the price process\n",
      "\n",
      "<h2><img src=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2015/03/RosenbaumPlot2.png\" align = \"center\" width=600></h2>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Estimating $\\eta$\n",
      "\n",
      "-   If we knew $\\eta$, we could use [(8)](#eq:efficientPrice):\n",
      "\n",
      "    $$X_{\\tau_i}=p_{t_i}-\\alpha\\,\\left(\\frac{1}{2}-\\eta\\right)\\,\\mbox{sign} (p_{t_i}-p_{t_{i-1}})$$\n",
      "    \n",
      "    to impute the efficient price process and then compute integrated\n",
      "    variance as\n",
      "    \n",
      "    $${RV}= \\sum_i (X_{\\tau_i}- X_{\\tau_{i-1}})^2.$$\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Schematic of continuation and reversal (one tick case)\n",
      "\n",
      "<a name=\"fig:CR\"></a>\n",
      "<h2><img src=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2015/03/CR.png\" align = \"center\" width=550></h2>\n",
      "Figure 2: If there is a continuation (orange), the efficient price $X$ moves by $2$.  If there is a reversal (green), the efficient price moves by $-2\\,\\eta$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Martingale argument to estimate $\\eta$\n",
      "\n",
      "-   Assume $\\alpha = 1$. From [Figure 2](#fig:CR), if a price change is a continuation, $X$ must\n",
      "    have moved by $2$; if a reversal, $X$ must have moved by $-2\\,\\eta$.\n",
      "\n",
      "\n",
      "-   Denote the probabilities of a continuation by $C$ and of a reversal\n",
      "    by $R$. Then, because\n",
      "    \n",
      "    $$\\mathbb{E} [\\Delta X ] = C\\,2 - R\\,2\\,\\eta = 0$$\n",
      "    \n",
      "    we must have\n",
      "    \n",
      "    $$\\eta = \\frac C R.$$\n",
      "\n",
      "\n",
      "-   In the case where the price may only move by one tick, the estimator\n",
      "    $\n",
      "    \\hat \\eta = C / R \n",
      "    $ is a consistent estimator of $\\eta$.\n",
      "\n",
      "    -   We recognize this as Large\u2019s alternation estimator.\n",
      "\n",
      "\n",
      "-   In an easy extension of the argument, if $C_k$ and $R_k$ are the\n",
      "    probabilities of continuations and reversals of size $k$, we have\n",
      "    \n",
      "    $$\\mathbb{E} [\\Delta X ] = C\\,k - R\\,(k-1+2\\,\\eta) = 0.$$\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Estimation of $\\eta$ in general\n",
      "\n",
      "Let $\\alpha$ be the tick size, $k$ the size of a price change in ticks.\n",
      "Then \n",
      "\n",
      "$$\\hat \\eta = \\sum_{k=1}^T\\,\\lambda_k\\,u_k$$ \n",
      "\n",
      "with\n",
      "\n",
      "$$\\lambda_k=\\frac{C_k+R_k}{\\sum_{j=1}^m(C_j+R_j)};\\quad u_k=\\frac{1}{2}\\,\\left(k\\,\\left(\\frac{C_k}{R_k}-1\\right)+1\\right)$$\n",
      "\n",
      "where $C_k$ and $R_k$ are the numbers of continuations and reversals of size $k$.\n",
      "\n",
      "-   The UZ estimator can be viewed as a generalization of the $ALT$\n",
      "    estimator of Large to markets where the price may move by several\n",
      "    ticks.\n",
      "\n",
      "\n",
      "-   In fact, the UZ estimator can be made robust to bad data by\n",
      "    considering only moves of one tick.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Estimation of integrated variance\n",
      "\n",
      "Integrated variance is estimated as\n",
      "$$\\widehat {RV}= \\sum_i (\\hat X_{t_i}-\\hat X_{t_{i-1}})^2$$ with\n",
      "$$\\hat X_{t_i}=p_{t_i}-\\alpha\\,\\left(\\frac{1}{2}-\\hat \\eta\\right)\\,\\mbox{sign} (p_{t_i}-p_{t_{i-1}})$$\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Pros and cons of the Robert-Rosenbaum model\n",
      "\n",
      "-   Pros:\n",
      "\n",
      "    -   Prices quantized in ticks\n",
      "\n",
      "    -   Bid-ask bounce\n",
      "\n",
      "    -   Inverse relation between durations and volatility.\n",
      "\n",
      "    -   Price moves of several ticks are generated endogenously.\n",
      "\n",
      "    -   Autocorrelograms and cross correlograms are realistic, both in\n",
      "        calendar time and in trading time.\n",
      "\n",
      "\n",
      "-   Cons:\n",
      "\n",
      "    -   The data need to be very clean. If there are any outliers, they\n",
      "        badly affect the UZ volatility estimate.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Estimates of realized variance of BAC data\n",
      "\n",
      "To make the estimates easier to interpret, we quote\n",
      "$\\sqrt{RV} \\sim \\sigma\\,\\sqrt{T}$. Thus, a good estimate should be of\n",
      "the order of $2\\%$ or so.\n",
      "\n",
      "<span>Table 1. BAC data from 04-May-2012.</span>\n",
      "\n",
      "\n",
      "|   | Trades | Quotes |\n",
      "|---|--------|--------|\n",
      "| RV(q=1)  | 7.46%       |   1.75%     |\n",
      "| RV(q=229)$^1$ | 2.34%       |   2.19%     |\n",
      "| TSRV(q=5)  | 1.96%       |   2.01%     |\n",
      "| MSRV(q=5)  | 1.95%       |   2.01%     |\n",
      "| KRVTH(q=5)  | 1.94%       |   2.02%     |\n",
      "| KRVC(q=5)  | 1.94%       |   2.01%     |\n",
      "\n",
      "---\n",
      "$^1$This is roughly 5 minutes\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### BAC volume profile\n",
      "\n",
      "<span>BAC data from 04-May-2012.</span>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%R\n",
      "\n",
      "#----------------------------------------------------------------------------------\n",
      "# Compute and plot volume and realized variance profiles\n",
      "\n",
      "ep <- endpoints(tqBAC,'minutes',k=1)\n",
      "ne <- length(ep)\n",
      "ep <- ep[-ne]\n",
      "sz <- function(x){mean(as.numeric(x$SIZE))}\n",
      "szVec <- period.apply(tqBAC, INDEX=ep, FUN= sz)\n",
      "ns <- length(szVec)\n",
      "plot(szVec, ylab = \"# Shares\", main=NA)\n",
      "ks <- ksmooth(1:ns,szVec, bandwidth=50)$y\n",
      "sz2 <- as.xts(cbind(szVec,ks))\n",
      "names(sz2) <- c(\"SIZE\", \"kSIZE\")\n",
      "lines(sz2$kSIZE,col=\"red\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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aIx6f/v375+bmtsgCY4SNGzfqdLobbrghLS3N7XaPHDkSz5OkpKTWPhlOnDgB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\nEwRBqCmUoAmCINSU/wdwYVAVSV3S8AAAAABJRU5ErkJggg==\n"
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "-   <span>The red line is kernel-smoothed volume</span>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### BAC volume and realized variance profiles\n",
      "\n",
      "<span>BAC data from 04-May-2012.</span>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%R\n",
      "\n",
      "# Now compute realized variance from mid-quotes with the Tukey Hanning kernel (q=5) \n",
      "rv <- function(x){\n",
      "  mid <- (as.numeric(x$BID) + as.numeric(x$OFR))/2\n",
      "  rv <-KRVTH(mid,5)\n",
      "  return(rv)\n",
      "}\n",
      "sz <- function(x){mean(as.numeric(x$SIZE))}\n",
      "\n",
      "ep <- endpoints(tqBAC,'minutes',k=5)\n",
      "ep <- ep[-length(ep)]\n",
      "szVec <- period.apply(tqBAC, INDEX=ep, FUN= sz)\n",
      "rvVec <- period.apply(tqBAC, INDEX=ep, FUN= rv)\n",
      "#plot(rvVec)\n",
      "\n",
      "# Show the volume and RV plots together\n",
      "par(mar = c(5,5,2,5),cex=0.8)\n",
      "plot(szVec, ylab = \"# Shares\", main=NA)\n",
      "par(new = T)\n",
      "plot(rvVec, col = \"red\", axes = F, xlab = NA, ylab = NA,main=NA)\n",
      "axis(side = 4)\n",
      "mtext(side = 4, line = 3, \"RV\")\n",
      "legend(\"top\",inset=0.2,legend=c(\"Volume    \",\"RV\"),col=c(\"black\",\"red\"),lwd=c(1,1))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Error in axis(side = 4) : plot.new has not been called yet\n",
        "In addition: Warning message:\n",
        "In par(new = T) : calling par(new=TRUE) with no plot\n",
        "Error in axis(side = 4) : plot.new has not been called yet\n"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "-   <span>The red line is kernel-smoothed volume</span>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Estimating covariance\n",
      "\n",
      "-   The natural estimator of covariance is just\n",
      "\n",
      "    $$\\frac{1}{T}\\,\\sum_{i=1}^T\\,\\Delta p^{(1)}_i\\,\\Delta p^{(2)}_i$$\n",
      "    \n",
      "    where $p^{(1)}$ and $p^{(2)}$ are prices of two assets.\n",
      "\n",
      "\n",
      "-   However the prices $p^{(i)}$ are usually asynchronous.\n",
      "\n",
      "\n",
      "-   Consequently, this estimator decreases as sampling frequency\n",
      "    increases.\n",
      "\n",
      "    -   In fact, [Hayashi and Yoshida]<sup id=\"cite_ref-HayashiYoshida\" class=\"reference\"><a href=\"#cite_note-HayashiYoshida\"><span>[</span>4<span>]</span></a></sup> show under pretty innocuous assumptions\n",
      "        that it tends to zero as sampling frequency increases.\n",
      "\n",
      "\n",
      "-   This is called the <span>*Epps Effect*</span>.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### The Hayashi-Yoshida estimator\n",
      "\n",
      "$$HY_t=\\sum_{i \\leq t}\\,\\sum_{j \\leq t}\\,(p^{(1)}_i-p^{(1)}_{i-1})\\,(p^{(2)}_j-p^{(2)}_{j-1})\\,v_{ij}$$\n",
      "where\n",
      "\n",
      "$$v_{ij}={\\bf 1}_{[t_{i-1},t_i] \\cap [t_{j-1},t_j] \\neq \\emptyset}$$\n",
      "\n",
      "In\n",
      "words, the sum is over all overlapping intervals.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### How does Hayashi-Yoshida work?\n",
      "\n",
      "Suppose\n",
      "\n",
      "\\begin{eqnarray*}\n",
      "dp^{(1)}_t&=&\\sigma_1\\,dZ_1\\\\\n",
      "dp^{(2)}_t&=&\\sigma_2\\,dZ_2\\end{eqnarray*}\n",
      "\n",
      "with $\\mathbb{E} [dZ_1\\,dZ_2]=\\rho\\,dt$. Then\n",
      "\n",
      "\\begin{eqnarray*}\n",
      "\\mathbb{E} \\left[(p^{(1)}_i-p^{(1)}_{i-1})\\,(p^{(2)}_j-p^{(2)}_{j-1})\\right]\n",
      "&=&\\mathbb{E} \\left[\\int_{t_{i-1}}^{t_i}\\,\\sigma_1\\,dZ_1\\,\\int_{t_{j-1}}^{t_j}\\,\\sigma_2\\,dZ_2\\right] \n",
      "= \\int_{[t_{i-1},t_i] \\cap [t_{j-1},t_j] }\\,\\rho\\,\\sigma_1\\,\\sigma_2\\,dt.\n",
      "\\end{eqnarray*}\n",
      "\n",
      "Summing over all partitions of the interval $[0,T]$ gives the result."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### The Corsi HAR-RV model\n",
      "\n",
      "-   The package highfrequency implements a regression to\n",
      "    fit the parameters of the so-called HAR-RV (Heterogeneous\n",
      "    Autoregressive Realized Variance) model of [Corsi]<sup id=\"cite_ref-Corsi\" class=\"reference\"><a href=\"#cite_note-Corsi\"><span>[</span>2<span>]</span></a></sup>.\n",
      "\n",
      "\n",
      "-   This model can be regarded as a poor man\u2019s version of a long memory\n",
      "    model such as ARFIMA.\n",
      "\n",
      "    -   True long-memory models such as ARFIMA are notoriously hard to\n",
      "        fit.\n",
      "\n",
      "\n",
      "-   HAR-RV can also be considered an intelligent alternative to GARCH.\n",
      "\n",
      "\n",
      "-   The model boils down to the regression\n",
      "\n",
      "    $$RV_{t,t+h} = \\beta_0 + \\beta_D\\,RV_t + \\beta_W\\,RV_{t-5,t} + \\beta_M\\,RV_{t-22,t} + \\epsilon_{t,t+h}.$$\n",
      "    \n",
      "    In words, the RV forecast for $h$ days from now is a linear\n",
      "    combination of the current realized variance and (aggregate) RV\n",
      "    estimates for the last week and the last month."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Example "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%R \n",
      "# Forecasting daily Realized volatility for DJI 2008 using the basic harModel: HARRV\n",
      "data(realized_library); #Get sample daily Realized Volatility data\n",
      "DJI_RV <- realized_library$Dow.Jones.Industrials.Realized.Variance; #Select DJI\n",
      "DJI_RV <- DJI_RV[!is.na(DJI_RV)] #Remove NA's\n",
      "DJI_RV <- DJI_RV['2008']\n",
      "\n",
      "x <- harModel(data=DJI_RV , periods = c(1,5,22), RVest = c(\"rCov\"), type=\"HARRV\",h=1,transform=NULL)\n",
      "plot(x)\n",
      "print(summary(x))\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Error in get(\".xts_chob\", .plotxtsEnv) : object '.xts_chob' not found\n"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### RV data from the Oxford-Man Institute\n",
      "\n",
      "-   In principle, it is straightforward to compute RV every day for any\n",
      "    given underlying using one of the better RV estimators we presented\n",
      "    above.\n",
      "\n",
      "\n",
      "-   In practice, this involves a lot of data work, especially cleaning.\n",
      "\n",
      "\n",
      "-   The R-package highfrequency has some historical RV\n",
      "    data sourced from the Oxford-Man Institute of Quantitative Finance\n",
      "    Realized Library.\n",
      "\n",
      "    -   More such data is publicly available at\n",
      "        <http://realized.oxford-man.ox.ac.uk>.\n",
      "\n",
      "        -   In particular, various historical RV estimates for SPX are\n",
      "            available.\n",
      "\n",
      "        -   Data is updated daily.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Summary\n",
      "\n",
      "-   There has been a huge expansion in the literature on realized\n",
      "    variance and covariance estimation since around 2003 with many very\n",
      "    interesting papers.\n",
      "\n",
      "\n",
      "-   As a result, we now have very efficient estimators for realized\n",
      "    variance that take into account all of the available information.\n",
      "\n",
      "    -   The newer volatility estimators are all very much more efficient\n",
      "        that RV sampled every 5 minutes.\n",
      "\n",
      "    -   Moreover, kernel-based estimators are easily updated in real\n",
      "        time by adding the most recent tick and dropping the oldest\n",
      "        tick.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### References\n",
      "<br />\n",
      "\n",
      "<div class=\"reflist\" style=\"list-style-type: decimal;\">\n",
      "\n",
      "<ol>\n",
      "\n",
      "  <li id=\"cite_note-BNS\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-BNS\">^</a></b></span>O.E Barndorff-Nielsen, P.R Hansen, A Lunde, N\n",
      "Shephard, Realized kernels in practice: Trades and quotes,\n",
      "<span>*Econometrics Journal*</span> <span>**12 (3)**</span> 1\u201332 (2009).\n",
      "  </li>\n",
      "  \n",
      "  <li id=\"cite_note-Corsi\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-Corsi\">^</a></b></span>Fulvio Corsi, A simple approximate long-memory model of realized\n",
      "volatility, <span>*Journal of Financial Econometrics*</span>\n",
      "<span>**7**</span>(2) 174\u2013196 (2009).\n",
      "  </li>\n",
      "  \n",
      "  <li id=\"cite_note-GatheralOomen\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-GatheralOomen\">^</a></b></span>Jim Gatheral and Roel C. A Oomen, Zero-intelligence\n",
      "realized variance estimation, <span>*Finance and Stochastics\n",
      "*</span><span>**14**</span>(2) 249\u2013283 (2010).\n",
      "  </li>\n",
      "  <li id=\"cite_note-HayashiYoshida\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-HayashiYoshida\">^</a></b></span>Takaki Hayashi and Nakahiro Yoshida, On covariance\n",
      "estimation of non-synchronously observed diffusion processes,\n",
      "<span>*Bernoulli *</span><span>**11**</span>(2) 359\u2013379 (2005).\n",
      "  </li>\n",
      "  \n",
      "    <li id=\"cite_note-McAleerMedeiros\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-McAleerMedeiros\">^</a></b></span>Michael McAleer and Marcelo C. Medeiros ,\n",
      "Realized Volatility: A Review, <span>*Econometric Reviews*</span>\n",
      "<span>**27**</span>(1) 10\u201345 (2008).\n",
      "  </li>\n",
      "  \n",
      "    <li id=\"cite_note-RobertRosenbaum\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-RobertRosenbaum\">^</a></b></span>Christian Y. Robert and Mathieu Rosenbaum,\n",
      "Volatility and covariation estimation when microstructure noise and\n",
      "trading times are endogenous, <span>*Mathematical Finance*</span>\n",
      "<span>**22**</span>(1), 133\u2013164 (2012).\n",
      "  </li>\n",
      "</ol>"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}